Skip to main content
Journal of the Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society Cover Image
Download PDF

Subordination and inclusion theorems for higher order derivatives of a generalized fractional differintegral operator

Journal of the Egyptian Mathematical Society volume 27, Article number: 17 (2019) Cite this article

Abstract

The main object of this paper is to investigate some subordination results of certain subclasses of multivalent analytic functions which are defined by a generalized fractional differintegral operator. Inclusion relations for functions in the class \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \) and the images of these functions by the generalized Bernardi-Libera-Livingston integral operator are also considered.

Introduction

Denote the class consisting of analytic and multivalent functions in the open unit disc \(\mathbb {U}=\{z\in \mathbb {C}:\left \vert z\right \vert <1\}\) of the form:

$$ f(z)=z^{p}+\sum_{n=1}^{\infty}a_{p+n}z^{p+n}\;(p\in\mathbb{N}=\{1,2,...\}), $$
(1)

by \(\mathcal {A}(p).\) We note that \(\mathcal {A}(1)=\mathcal {A}.\)

Consider the first-order differential subordination

$$H(\varphi(z),z\varphi^{\prime}(z))\prec h(z), $$

where the symbol ≺ stands for subordination of two analytic functions in \(\mathbb {U\ }\)(see [1, 2]). A univalent function q is called dominant, if φ(z)≺q(z) for all analytic functions φ that satisfy this differential subordination. A dominant \(\widetilde {q}\) is called the best dominant, if \(\widetilde {q}(z)\prec q(z)\) for all dominant q. For \(f\in \mathcal {A}(p)\), the qth order derivative of f(z) could be written as

$$\begin{array}{*{20}l} {}f^{(q)}(z) & =\delta(p,q)z^{p-q}+\sum_{n=1}^{\infty}\delta(p+n,q)a_{p+n} z^{p+n-q},\;z\in\mathbb{U} \left(p>q,\;q\in\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}\right), \end{array} $$
(2)

where

$$\delta(p,q)=\frac{p!}{(p-q)!}:=\left\{ \begin{array} [c]{lll} p(p-1)\dots(p-q+1), & \text{if} & q\neq0,\\ 1, & \text{if} & q=0. \end{array} \right. $$

Let

$$ {\!~\!}_{p}F_{q}\left(a_{1},...,a_{p};b_{1},...,b_{q};z\right) =\sum\limits_{n=0}^{\infty}\frac{(a_{1})_{n}...(a_{p})_{n}}{(b_{1})_{n}...(b_{q})_{n}(1)_{n}}z^{n}, $$
(3)

be the well-known generalized hypergeometric function for complex parameters \(a_{1},...,a_{q}\,,\ b_{1},...,b_{s}\ \ (b_{j}\notin \mathbb {Z}_{0}^{-}=\{0,-1,-2,...\};\ j=1,2,...,s)\ \)and (λ)ν is the Pochhammer symbol defined by

$$(\lambda)_{\upsilon}:=\left\{ \begin{array} [c]{lll} 1 & & \text{if}\ \upsilon=0,\\ \lambda(\lambda+1)(\lambda+2)...(\lambda+\upsilon-1) & & \text{if } \upsilon\in \mathbb{N}. \end{array} \right. $$

In addition, if we put p=2, q=1, a1=a, a2=b, b1=c in (3), we get the (Gaussian) hypergeometric function 2F1(a,b;c;z)(c≠0,−1,−2,…) which satisfies (see [3])

$$ {}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}dt=\tfrac{\Gamma(b)\Gamma (c-b)}{\Gamma(c)}\;_{2}F_{1}(a,b;c;z)\;(\mathfrak{R}(c)>\mathfrak{R}(b)>0); $$
(4)
$$ {\!~\!}_{2}F_{1}(a,b;c;z)={(1-z)^{-a}}_{\,2}F_{1}\left(a,c-b;c;\frac{z}{z-1}\right) ; $$
(5)

and

$$ {\!~\!}_{2}F_{1}(a,b;c;z){=}_{2}F_{1}(b,a;c;z). $$
(6)

We will recall some definitions which will be used in our paper.

Definition 1

[4–12]. Assume that 0≤λ<1 and \(\mu,\eta \in \mathbb {R}\). Then, in terms of 2F1, the generalized fractional derivative operator for \(f\in \mathcal {A}(p)\) is defined by

$${}J_{0,z}^{\lambda,\mu,\eta,p}f(z):=\frac{d}{dz}\left[ \frac{z^{\lambda-\mu} }{\Gamma(1-\lambda)}\int_{0}^{z}(z-\zeta)^{-\lambda}f(\zeta)_{2}F_{1}\left(\mu-\lambda,1-\eta;1-\lambda;1-\frac{\zeta}{z}\right) d\zeta\right], $$

where f is an analytic function in a simply-connected region of the complex z-plane containing the origin with the order f(z)=O(|z|ε), z→0 when ε> max{0,μ−η}−1 and the multiplicity of (z−ζ)−λ is removed by requiring log(z−ζ)to be real when z−ζ>0.

Remark 1

We note that

(i) \(J_{0,z}^{\lambda,\mu,\eta,p}\left \{ z^{p+n}\right \} =\frac {\Gamma (p+n+1)\Gamma (p+n+1-\mu +\eta)}{\Gamma (p+n+1-\mu)\Gamma (p+n+1-\lambda +\eta)}z^{p+n-\mu }\ \left (n\geq 1\right),\)

(ii) \(J_{0,z}^{\lambda,\lambda,\eta,p}f(z)=D_{z}^{\lambda }f(z)\ \)(see [13]).

Goyal and Prajapat [14] (see also [4–12]) defined the operator \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}:\mathcal {A}(p)\rightarrow \mathcal {A}(p) \left (0\leq \lambda <1,\;\mu < p+1,\;\eta >\max \{\lambda,\mu \}-p-1\right),\ \)by

$$\begin{array}{*{20}l} {}\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z) & :=\frac{\Gamma(p+1-\mu)\Gamma(p+1-\lambda+\eta)}{\Gamma(p+1)\Gamma(p+1-\mu+\eta)}z^{\mu} J_{0,z}^{\lambda,\mu,\eta,p}f(z)\\ & =z^{p}+\sum\limits_{n=1}^{\infty}\frac{(p+1)_{n}(p+1-\mu+\eta)_{n} }{(p+1-\mu)_{n}(p+1-\lambda+\eta)_{n}}a_{p+n}z^{p+n}\\ & ={z^{p}}_{3}F_{2}\left(1,p+1,p+1-\mu+\eta;p+1-\mu,p+1-\lambda +\eta;z\right) \ast f(z),\\ & \end{array} $$
(7)

where the symbol ∗ stands for convolution of two power series and \(f\in \mathcal {A}(p)\). It is easy to check that

$$ z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime} =(p-\mu)\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)+\mu\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z). $$
(8)

In this paper, we define the higher order derivative of \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\ \)as follows:

$$\begin{array}{*{20}l} {}\left(\!\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\!\right)^{\left(q\right) }\!\,=\,\delta\left(p,q\right) z^{p-q}\,+\,\sum\limits_{n=1}^{\infty}\!\frac{(p+1)_{n}(p+1-\mu+\eta)_{n}}{(p+1-\mu)_{n}(p+1-\lambda+\eta)_{n}} \delta\!\left(p+n,q\right) a_{p+n}z^{p+n-q}\\ \left(p\in \mathbb{N},\ q\in\mathbb{N}_{0},\ p>q,\ 0\leq\lambda<1,\;\mu< p+1,\;\eta>\max\{\lambda,\mu\}-p-1\right). \end{array} $$
(9)

From (9), we have

$$\begin{array}{*{20}l} z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right)} & =(p-\mu)\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta +1,p}f(z)\right)^{\left(q-1\right) }+\left(\mu-q+1\right) \\ & \quad\times\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }\ \left(q\in \mathbb{N} \right). \end{array} $$
(10)

We say that \(f\in \mathcal {A}(p)\) is in the class \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \)if

$$ \frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}-\zeta\right) \prec\frac{1+Az}{1+Bz}, $$
(11)

\(0\leq \lambda <1,\;\mu < p+1,\;\eta >\max \{\lambda,\mu \}-p-1,\ 0\leq \zeta < p-q,\ -1\leq B<A\leq 1,\ p\in \mathbb {N},\ q\in \mathbb {N}_{0}\ {and}\ p>q+\zeta.\ \)Denoting by \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right), \) the class of functions \(f\left (z\right) \in \mathcal {A}(p)\ \)which satisfies

$$ \mathfrak{R}\left\{ \frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}-\zeta\right) \right\} >\xi\;(\xi<1;\ p\in\mathbb{N};\ z\in\mathbb{U}). $$
(12)

Preliminaries

To prove our main results, we shall need the following definition and lemmas.

Definition 2

[2]. Denote the set of all functions f that are analytic and univalent on \(\overline {\mathbb {U}}\setminus E(f)\ \)by \(\mathcal {Q}\), where

$$E(f):=\{\varsigma\in\partial\mathbb{U}:\underset{z\rightarrow\varsigma}{\lim }f(z)=\infty\}, $$

and are such that f′(ς)≠0 for \(\varsigma \in \partial \mathbb {U}\setminus E(f)\).

Lemma 1

[15]. Let h(z) be analytic and convex (univalent) function in \(\mathbb {U}\) with h(0)=1. Also let ϕ given by

$$\phi(z)=1+c_{n}z^{n}+c_{n+1}z^{n+1}+... $$

be analytic in \(\mathbb {U}\). If

$$ \phi(z)+\frac{z\phi^{\prime}(z)}{\alpha}\prec h(z)\;(\mathfrak{R}(\alpha)\geqslant 0;\ \alpha\neq0), $$
(13)

then

$$\phi(z)\prec\psi(z)=\frac{\alpha}{n}z^{-\frac{\alpha}{n}}\int_{0}^{z} t^{\frac{\alpha}{n}-1}h(t)dt\prec h(z), $$

and ψ is the best dominant of (13).

Lemma 2

[16]. Let h be a convex functions with

$$\mathfrak{R}\left[ \beta h(z)+\gamma\right] >0\ \left(\beta,\gamma\in \mathbb{C},\ z\in\mathbb{U}\right). $$

If p(z) is analytic in \(\mathbb {U\ }{with}\ p(0)=h(0),\ \)then

$$p(z)+\frac{zp^{\prime}\left(z\right) }{\beta p(z)+\gamma}\prec h(z)\Rightarrow p(z)\prec h(z). $$

The class of star-like (and normalized) functions of order α in \(\mathbb {U},\ \)is

$$\mathcal{S}^{\ast}\left(\alpha\right) =\left\{ f\in\mathcal{A}:\mathfrak{R}\left(\frac{zf^{\prime}\left(z\right) }{f\left(z\right) }\right) >\alpha\ \left(\alpha<1;\ z\in\mathbb{U}\right) \right\}. $$

Also in [17], if β>0 and β+γ>0, for a given \(\alpha \in \left [ -\frac {\gamma }{\beta },1\right),\) we define the order of starlikeness of the class \(\mathrm {I}_{\beta,\gamma }\left [ \mathcal {S}^{\ast }\left (\alpha \right) \right ]\ \)by the largest number 𝜗(α;β,γ) such that\(\ \mathrm {I}_{\beta,\gamma }\left [ \mathcal {S}^{\ast }\left (\alpha \right) \right ] \subset \mathcal {S}^{\ast }\left (\vartheta \right),\ \)where Iβ,γ is given by

$$ \mathrm{I}_{\beta,\gamma}(f)(z)=\left[ \frac{\beta+\gamma}{z^{\gamma}} \int\limits_{0}^{z}f^{\beta}(t)t^{\gamma-1}dt\right]^{\frac{1}{\beta}}. $$
(14)

Lemma 3

[17]. Let β>0, β+γ>0 and consider Iβ,γ defined by (14). If \(\alpha \in \left [-\frac {\gamma }{\beta },1\right),\ \)then the order of starlikeness of the class \(\mathrm {I}_{\beta,\gamma }\left [ \mathcal {S}^{\ast }\left (\alpha \right) \right ] \ \)is given by the number \(\vartheta \left (\alpha ;\beta,\gamma \right) =\inf \left \{ \mathfrak {R}\left (q(z)\right):z\in \mathbb {U}\right \},\ \)where

$$q(z)=\frac{1}{\beta Q\left(z\right) }-\frac{\gamma}{\beta}\ \text{and\ } Q(z)=\int\limits_{0}^{1}\left(\frac{1-z}{1-tz}\right)^{2\beta\left(1-\alpha\right) }t^{\beta+\gamma-1}dt. $$

Moreover, if \(\alpha \in \left [ \alpha _{0},1\right),\ {where}\ \alpha _{0}=\max \left \{ \frac {\beta -\gamma -1}{2\beta };-\frac {\gamma }{\beta }\right \}\ {and}\ g=\mathrm {I}_{\beta,\gamma }\left (f\right)\) with \(f\in \mathcal {S}^{\ast }\left (\alpha \right),\ \)then

$$\mathfrak{R}\left(\frac{zg^{\prime}\left(z\right) }{g\left(z\right) }\right) >\vartheta\left(\alpha;\beta,\gamma\right) \ \left(z\in\mathbb{U}\right), $$

where

$$\vartheta\left(\alpha;\beta,\gamma\right) =\frac{1}{\beta}\left[\frac{\beta+\gamma}{{\!~\!}_{2}F_{1}\left(1,2\beta\left(1-\alpha\right),\beta+\gamma+1;\frac{1}{2}\right) }-\gamma\right]. $$

Subordination and Inclusion theorems involving \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left (q\right) }\)

We assume throughout this paper unless otherwise mentioned that\(\ p\in \mathbb {N},\ 0\leq \lambda <1,\;\mu < p,\;\eta >\max \{\lambda,\mu \}-p-1,\ -1\leq B<A\leq 1,\ 0\leq \zeta < p-q,\ \xi <1,\ \sigma >0,\ 0< c\leq 1\) and the powers are considered principal ones.

Theorem 1

Assume that \(1\leq q\leq p\ {and}\ f\left (z\right)\in \mathcal {A}\left (p\right) \) satisfy

$$(1-\sigma)\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}+\sigma \frac{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\prec \sqrt{1+cz}, $$

then

$$ \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\prec Q(z)\prec \sqrt{1+cz}, $$
(15)

where

$$ Q(z)={\left(1+cz\right)^{\frac{1}{2}}}_{\,2}F_{1}\left(-\frac{1} {2},1;\frac{p-\mu}{\sigma}+1;\frac{cz}{1+cz}\right), $$
(16)

is the best dominant of (15). Furthermore,

$$ \mathfrak{R}\left\{ \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\right\} >M, $$
(17)

where

$$M={\left(1-c\right)^{\frac{1}{2}}}_{\,2}F_{1}\left(-\frac{1}{2},1;\frac{p-\mu}{\sigma}+1;\frac{c}{c-1}\right). $$

The estimate in (17) is the best possible.

Proof

Putting

$$ \phi(z)=\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\ (z\in \mathbb{U}), $$
(18)

then ϕ(z) is analytic in \(\mathbb {U}\). After some computations, we get

$$(1-\sigma)\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}+\sigma \frac{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}} $$
$$=\phi(z)+\left(\frac{\sigma}{p-\mu}\right) z\phi^{\prime}(z)\prec \sqrt{1+cz}. $$

where the influence of \(h(z) = \sqrt {1+cz}\) under certain values of c is illustrated by Fig. 1. To apply Lemma 1, it suffies to show that h(z) is convex, therefore for z=reiθ, r∈(0,1), θ∈[−π,π], we have

$$1+\frac{zh^{\prime\prime}}{h^{\prime}}=1-\frac{cz}{2\left(1+cz\right) }=\frac{2+cz}{2\left(1+cz\right) }, $$

and

$$\begin{array}{*{20}l} \mathfrak{R}\left(1+\frac{zh^{\prime\prime}}{h^{\prime}}\right) & =\frac{2+3cr\cos\theta+c^{2}r^{2}}{\left\vert 1+cre^{i\theta}\right\vert^{2}}\geq\frac{2-3cr+c^{2}r^{2}}{\left\vert 1+cre^{i\theta}\right\vert^{2}}\\ & =\frac{\left(2-cr\right) \left(1-cr\right) }{\left\vert 1+cre^{i\theta}\right\vert^{2}}>0. \end{array} $$
Fig. 1
figure1

Influence of \(h(z)=\sqrt {1+cz}\) for different values of c

Full size image

This implies that h is convex in \(\mathbb {U}\).

Now, by using Lemma 1 (with n=1) and making a change of variables followed by the use of (4) and (5), we deduce that

$$\begin{array}{*{20}l} \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}} & \prec Q(z)=\frac {p-\mu}{\sigma}z^{-\frac{p-\mu}{\sigma}}\int_{0}^{z}t^{\frac{p-\mu}{\sigma} -1}\left(1+ct\right)^{\frac{1}{2}}dt\\ & ={\left(1+cz\right)^{\frac{1}{2}}}_{\,2}F_{1}\left(-\frac{1}{2},1;\frac{p-\mu}{\sigma}+1;\frac{cz}{1+cz}\right), \end{array} $$

this proves (15). Next, it is enough to show that

$$\inf_{\left\vert z\right\vert <1}\left\{ \mathfrak{R}(Q(z)\right\} =Q(-1). $$

Indeed

$$\mathfrak{R}\left\{ \sqrt{1+cz}\right\} \geqslant\sqrt{1-cr}\ \left(\left\vert z\right\vert \leq r<1\right). $$

Setting

$$G(z,s)=\sqrt{1+czs}\;\text{and\ }d\nu(s)=\frac{p-\mu}{\sigma}s^{\frac{p-\mu }{\sigma}-1}ds\;(0\leq s\leq1), $$

which is a positive measure on the closed interval [0,1], we get

$$Q(z)=\int\limits_{0}^{1}G(z,s)d\nu(s), $$

so that

$$\mathfrak{R}\left\{ Q(z)\right\} \geqslant\int_{0}^{1}\sqrt{1-cr}d\nu(s)=Q(-r)\;(\left\vert z\right\vert \leq r<1). $$

Letting r→1− in the above inequality, we obtain (17). To show that the result in (17) is sharp, let us suppose that

$$\mathfrak{R}\left\{ \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\right\} >M_{1}, $$

that is

$$\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\prec\frac{1+\left(1-2M_{1}\right) z}{1-z}. $$

From (15), we have

$$\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\prec\frac{1+\left(1-2M\right) z}{1-z}, $$

and so

$$\frac{1+\left(1-2M\right) z}{1-z}\prec\frac{1+\left(1-2M_{1}\right) z}{1-z}, $$

which implies that M≤M1, that is, M cannot be decreased and the estimate in (17) is the best possible. □

For \(f\in \mathcal {A}(p)\) the generalized Bernardi-Libera-Livingston integeral operator Fp,υ is defined by (see [18]):

$$\begin{array}{*{20}l} F_{p,\upsilon}f(z) & =\frac{\upsilon+p}{z^{p}}\int\limits_{0}^{z} t^{\upsilon-1}f(t)dt\\ & =\left(z^{p}+\sum\limits_{n=1}^{\infty}\frac{\upsilon+p}{\upsilon +p+n}z^{p+n}\right) \ast f(z)\\ & ={z^{p}}_{\,3}F_{2}\left(1,1,\upsilon+p;1,\upsilon+p+1;z\right) \ast f(z)\ (\upsilon>-p). \end{array} $$
(19)

Lemma 4

If \(f\in \mathcal {A}\left (p\right),\ \)then (i) \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}\left (F_{p,\upsilon }f\right) =F_{p,\upsilon }\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f\right),\)

(ii)

$$ z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\prime}=(p+\upsilon)\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)-\upsilon \mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z), $$
(20)

(iii)

$$ {}z\left(\! \mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\!\right)^{\left(q\right) }\!=(p+\upsilon)\left(\! \mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\!\right)^{\left(q-1\right) }-\left(\upsilon+q-1\right) \left(\! \mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\!\right)^{\left(q-1\right) }. $$
(21)

Proof

Since \(f(z)\in \mathcal {A}(p)\), then

$$\begin{array}{*{20}l} {}\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}\left(F_{p,\upsilon}f\right) &\,=\,\left[ {z^{p}}_{3}F_{2}\left(1,p+1,p+1-\mu+\eta;p+1-\mu,p+1-\lambda +\eta;z\right) \right] \ast\left(F_{p,\upsilon}f\right) \\ & \,=\,\left[ {z^{p}}_{3}F_{2}\left(1,p+1,p+1-\mu+\eta;p+1-\mu,p+1-\lambda +\eta;z\right) \right] \\ &\quad\ast \left[ {z^{p}}_{3}F_{2}\left(1,1,\upsilon+p;1,\upsilon+p+1;z\right) \ast f(z)\right], \end{array} $$

and

$$\begin{array}{*{20}l} {}F_{p,\upsilon}\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f\right) &={z^{p}}_{3}F_{2}\left(1,1,\upsilon+p;1,\upsilon+p+1;z\right) \ast\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f\right) \\ & ={z^{p}}_{3}F_{2}\left(1,1,\upsilon+p;1,\upsilon+p+1;z\right) \ast\\ & \left[ {z^{p}}_{3}F_{2}\left(1,p+1,p+1-\mu+\eta;p+1-\mu,p+1-\lambda+\eta;z\right) \ast f(z)\right]. \end{array} $$

Now, the first part of this lemma follows. Also, the recurrence relation of Fp,υ is given by

$$ z\left(F_{p,\upsilon}f(z)\right)^{\prime}=(p+\upsilon)f(z)-\upsilon F_{p,\upsilon}f(z). $$
(22)

If we replace f(z) by \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\ \)and using the first part of this lemma, we get (20). If we differentiate (20) q-times, we obtain (21). □

Theorem 2

Suppose that \(1\leq q\leq p\ {and}\ f\left (z\right)\in \mathcal {A}(p)\) satisfy

$$(1-\sigma)\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon }f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1} }+\sigma\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\prec \sqrt{1+cz}, $$

where Fp,υ defined by (19), then

$$ \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\prec \varphi(z)\prec\sqrt{1+cz}, $$
(23)

where φ(z) given by

$$\varphi(z)={\left(1+cz\right)^{\frac{1}{2}}}_{\,2}F_{1}\left(-\frac{1} {2},1;\frac{\upsilon+p}{\sigma}+1;\frac{cz}{1+cz}\right), $$

is the best dominant of (23). Further,

$$ \mathfrak{R}\left\{ \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon }f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1} }\right\} >L, $$
(24)

where

$$L={\left(1-c\right)^{\frac{1}{2}}}_{\,2}F_{1}\left(-\frac{1}{2},1;\frac{\upsilon+p}{\sigma}+1;\frac{c}{c-1}\right). $$

The result is the best possible.

Proof

Taking

$$ \Theta(z)=\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\ (z\in\mathbb{U)}, $$
(25)

then Θ is analytic in \(\mathbb {U}\). After some calculations, we have

$$\begin{array}{*{20}l} & (1-\sigma)\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}+\sigma\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q-1\right) }}{\delta\left(p,q-1\right) z^{p-q+1}}\\ & =\Theta(z)+\left(\frac{\sigma}{p+\upsilon}\right) z\Theta^{\prime }\left(z\right) \prec\sqrt{1+cz}. \end{array} $$

By employing the same technique that was used in proving Theorem 1, the remaining part of the theorem can be proved. □

Theorem 3

Let\(\ q\in \mathbb {N}_{0}\ {and}\ p>q+\zeta.\ \)If \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right),\ \)then \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta,\xi \right) \ \)for |z|<R(p,q,μ,ζ,ξ) where

$$ R\left(p,q,\mu,\zeta,\xi\right) =\min\left\{ r>0:t\left(r\right) =0\right\}, $$
(26)

and

$$t\left(r\right) =1-\frac{2r}{\left(p-q-\zeta\right) \left\vert \left(1-\xi\right) \left(1-r\right)^{2}-\left\vert \xi+\frac{q+\zeta-\mu }{p-q-\zeta}\right\vert \left(1-r^{2}\right) \right\vert }. $$

Proof

Assume that \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right) \) and

$$ u\left(z\right) =\frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}-\zeta\right), $$
(27)

then, u(z) is analytic in \(\mathbb {U}\) with \(u(0)=1,\ \mathfrak {R}\left \{ u\left (z\right) \right \} >\xi.\) After some computations, we have

$$ {}u\left(z\right) +\frac{zu^{\prime}\left(z\right) }{\left(p-q-\zeta\right) u(z)+\left(q+\zeta-\mu\right) }\,=\,\frac{1}{p-q-\zeta}\!\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q\right) }}-\zeta\right). $$
(28)

Letting \(v(z)=\frac {u(z)-\xi }{1-\xi },\ \)then, v(0)=1 with \(\mathfrak {R}\left \{ v\left (z\right) \right \} >0.\ \)Substituting in (28), we obtain

$$\begin{array}{*{20}l} & \frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q\right) }}-\zeta\right) -\xi\\ & =\left(1-\xi\right) \left[ v\left(z\right) +\frac{zv^{\prime}\left(z\right) }{\left(p-q-\zeta\right) \left[ \left(1-\xi\right) v(z)+\xi\right] +\left(q+\zeta-\mu\right) }\right], \end{array} $$

and so

$$\begin{array}{*{20}l} & \mathfrak{R}\left\{ \frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q\right)}}-\zeta\right) -\xi\right\} \\ & \geq\left(1-\xi\right) \ \left[ \mathfrak{R}\left\{ v\left(z\right)\right\} -\frac{\left\vert zv^{\prime}\left(z\right) \right\vert }{\left(p-q-\zeta\right) \left\vert \left(1-\xi\right) \left\vert v\left(z\right) \right\vert -\left\vert \xi+\frac{q+\zeta-\mu}{p-q-\zeta}\right\vert\, \,\right\vert }\right] \\ & \geq\left(1-\xi\right) \ \left[ \mathfrak{R}\left\{ v\left(z\right)\right\} -\frac{\left\vert zv^{\prime}\left(z\right) \right\vert }{\left(p-q-\zeta\right) \left\vert \left(1-\xi\right) \ \mathfrak{R}\left\{ v\left(z\right) \right\} -\left\vert \xi+\frac{q+\zeta-\mu}{p-q-\zeta}\right\vert\, \,\right\vert }\right]. \end{array} $$

Applying the following well-known estimate [19]:

$$\mathfrak{R}\left\{ v(z)\right\} \geq\frac{1-r}{1+r}\ \text{and\ }\frac{\left\vert zv^{\prime}(z)\right\vert }{\mathfrak{R}\left\{ v(z)\right\} }\leq\frac{2nr^{n} }{1-r^{2n}}\ (\left\vert z\right\vert =r<1), $$

for n=1, we get

$$\mathfrak{R}\left\{ \frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q\right)}}-\zeta\right) -\xi\right\} \geq\left(1-\xi\right) t\left(r\right)\mathfrak{R}\left\{ v(z)\right\}. $$

It is easily seen that t(r) is positive, if |z|<R(p,q,μ,ζ,ξ), where R is given by (26). □

Theorem 4

Let \(f(z)\in \mathcal {A}(p),\ p>\mu,\ \gamma >0\ \)and

$$ \mathfrak{R}\left(\frac{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\prime}}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}-\frac{\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta +1,p}f(z)}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)}\right) <\frac{\gamma}{p-\mu}, $$
(29)

then

$$\mathfrak{R}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)}\right)^{-\frac{1}{2\gamma}}>\frac{1}{2}. $$

The result is sharp.

Proof

From (8), (29) may be written as

$$\mathfrak{R}\left(1+\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime\prime}}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}-\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)}\right) <\gamma, $$

or equivalently,

$$ 1+\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime\prime}}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}-\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)}\prec-\frac{2\gamma z}{1-z}. $$
(30)

Letting

$$\digamma(z)=\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)}\right)^{-\frac{1}{2\gamma}}, $$

then, we can express (30) as

$$ z\left(\log\digamma(z)\right)^{\prime}\prec z\left(\log\frac{1} {1-z}\right)^{\prime}. $$
(31)

Fom [20], (31) implies to

$$\digamma(z)\prec\frac{1}{1-z}, $$

or equivalently,

$$\mathfrak{R}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\prime}}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)}\right)^{-\frac {1}{2\gamma}}>\frac{1}{2}\ \left(z\in\mathbb{U}\right). $$

To show that the result is sharp, let

$$K(z)=z^{p}+\sum\limits_{n=1}^{\infty}\frac{(p+1)_{n}(p+1-\mu+\eta)_{n} }{(p+1-\mu)_{n}(p+1-\lambda+\eta)_{n}}\frac{2\gamma\left(2\gamma-1\right)...\left(2\gamma-n+1\right) }{n!}z^{p+n}, $$

and so

$$\begin{array}{*{20}l} \mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}K(z) & =z^{p}+\sum\limits_{n=1} ^{\infty}\frac{2\gamma\left(2\gamma-1\right)...\left(2\gamma-n+1\right) }{n!}z^{p+n}\\ & =z^{p}\left(1+z\right)^{2\gamma}. \end{array} $$

It is easy to check that K(z) satisfies (29) and

$$\mathfrak{R}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}K(z)\right)^{\prime}}{\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}K(z)}\right)^{-\frac {1}{2\gamma}}\rightarrow\frac{1}{2} $$

as z→1−. This ends our proof. □

Theorem 5

Consider that\(\ q\in \mathbb {N}_{0},\ p>q+\zeta \ {and}\ \)

$$ \left(p-q-\zeta\right) \left(1-A\right) +\left(q+\zeta-\mu\right) \left(1-B\right) \geq0. $$
(32)

(i) Suppose that \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast }:=\mathbb {U}\backslash \{0\},\ \)then

$$\mathcal{S}_{p,q}^{\lambda+1,\mu+1,\eta+1}\left(\zeta;A,B\right) \subset\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;A,B\right). $$

(ii) Also, assuming that

$$ \frac{1-A}{1-B}\geq\frac{1}{p-q-\zeta}\max\left\{ \frac{p-2q-2\zeta+\mu-1} {2},-\left(q+\zeta-\mu\right) \right\}, $$
(33)

then

$$\mathcal{S}_{p,q}^{\lambda+1,\mu+1,\eta+1}\left(\zeta;A,B\right) \subset\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta,\xi\right). $$

where the bound

$$ \xi\left(A,B\right) =\frac{1}{p-q-\zeta}\left[ \frac{p-\mu}{{\!~\!}_{2} F_{1}\left(1,\frac{2\left(p-q-\zeta\right) \left(A-B\right) } {1-B},p-\mu+1;\frac{1}{2}\right) }-\left(q+\zeta-\mu\right) \right], $$
(34)

is the best possible

Proof

Let \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta ;A,B\right) \ \)and

$$ G\left(z\right) =z\left(\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}{\delta\left(p,q\right) z^{p-q} }\right)^{\frac{1}{p-q-\zeta}}. $$
(35)

Since \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast },\ \) then G(z) is analytic in \(\mathbb {U}\) with G(0)=0 and G′(0)=1. Differentiating both sides of (35) logarithmically, we get

$$ \Psi\left(z\right) =\frac{zG^{\prime}\left(z\right) }{G\left(z\right)}=\frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}-\zeta\right). $$
(36)

Using (10) in (36), we have

$$ \left(p-\mu\right) \frac{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}=\left(p-q-\zeta\right) \Psi\left(z\right) +\left(q+\zeta-\mu\right). $$
(37)

Differentiating both sides of (37) logarithmically, we get

$${}\frac{1}{p-q-\zeta}\!\left(\!\frac{z\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda+1,\mu+1,\eta+1,p}f(z)\right)^{\left(q\right) }}\,-\,\zeta\!\right)\! \!=\Psi\left(z\right) +\frac{z\Psi^{\prime}\left(z\right)}{\left(p-q-\zeta\right) \Psi\left(z\right) +\left(q+\zeta-\mu\right)}. $$

Combining this identity together with \(f\left (z\right)\in \mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta ;A,B\right),\ \)we obtain

$$\Psi\left(z\right) +\frac{z\Psi^{\prime}\left(z\right) }{\left(p-q-\zeta\right) \Psi\left(z\right) +\left(q+\zeta-\mu\right) } \prec\frac{1+Az}{1+Bz}\equiv h(z). $$

We will use Lemma 2 for \(\widetilde {\beta }=\left (p-q-\zeta \right),\ \widetilde {\gamma }=\left (q+\zeta -\mu \right).\ \)Since h(z) is a convex function in \(\mathbb {U\ }\)and

$$\mathfrak{R}\left[ \left(p-q-\zeta\right) \frac{1+Az}{1+Bz}+\left(q+\zeta -\mu\right) \right] >0, $$

whenever (32) holds. Then \(f(z)\in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \)from Lemma 2. This completes the proof of (i). To prove (ii), we assume that (33) holds, then all the assumptions of Lemma 3 are satisfied for the above values of \(\widetilde {\beta },\ \widetilde {\gamma }\ \)and \(\widetilde {\alpha }=\frac {1-A}{1-B}.\ \)It follows that \(\mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta ;A,B\right) \subset \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right) \) where ξ(A,B) given by (34) is the best possible. □

Theorem 6

Assume that\(\ q\in \mathbb {N}_{0},\ p>q+\zeta \ {and}\ \)

$$ \left(p-q-\zeta\right) \left(1-A\right) +\left(q+\zeta+\upsilon\right) \left(1-B\right) \geq0. $$
(38)

(i) Suppose that \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon }f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast },\ \)then

$$\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;A,B\right) \subset F_{p,\upsilon}\left(\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;A,B\right) \right). $$

(ii) Also, assuming that

$$ \frac{1-A}{1-B}\geq\frac{1}{p-q-\zeta}\max\left\{ -\frac{q+2\zeta+\upsilon +1}{2},-\left(p+\zeta+\upsilon\right) \right\}, $$
(39)

then

$$\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;A,B\right) \subset \mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;\tau\left(A,B\right) \right). $$

where the bound

$$ {}\tau\left(A,B\right) =\frac{1}{p-q-\zeta}\left[ \frac{2p-q+\upsilon}{{\!~\!}_{2}F_{1}\left(1,\frac{2\left(p-q-\zeta\right) \left(A-B\right)}{1-B},2p-q+\upsilon;\frac{1}{2}\right) }-\left(p+\zeta+\upsilon\right) \right], $$
(40)

is the best possible.

Proof

Let \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \)and

$$ H\left(z\right) =z\left(\frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q\right) }}{\delta\left(p,q\right) z^{p-q}}\right)^{\frac{1}{p-q-\zeta}}. $$
(41)

Since \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon }f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast },\ \)then H(z) is analytic in \(\mathbb {U}\) with H(0)=0 and H′(0)=1. Differentiating both sides of (41) logarithmically, we get

$$ \Phi\left(z\right) =\frac{zH^{\prime}\left(z\right) }{H\left(z\right)}=\frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q\right) }}-\zeta\right). $$
(42)

Using (21) in (42), we have

$$ \left(p+\upsilon\right) \frac{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon}f(z)\right)^{\left(q\right) }}=\left(p-q-\zeta\right) \Phi\left(z\right) +\left(q+\zeta+\upsilon\right). $$
(43)

Differentiating both sides of (43) logarithmically, we get

$${}\frac{1}{p-q-\zeta}\left(\frac{z\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q+1\right) }}{\left(\mathcal{M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left(q\right) }}-\zeta\right) =\Phi\left(z\right) +\frac{z\Phi^{\prime}\left(z\right) }{\left(p-q-\zeta\right) \Phi\left(z\right) +\left(q+\zeta+\upsilon\right) }. $$

Combining this identity together with \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right),\ \)we obtain

$$\Phi\left(z\right) +\frac{z\Phi^{\prime}\left(z\right) }{\left(p-q-\zeta\right) \Phi\left(z\right) +\left(q+\zeta+\upsilon\right) }\prec\frac{1+Az}{1+Bz}\equiv h(z). $$

We will use Lemma 2 for \(\widetilde {\beta }=\left (p-q-\zeta \right),\ \overline {\gamma }=\left (q+\zeta +\upsilon \right).\ \)Since h(z) is a convex function in \(\mathbb {U\ }\)and

$$\mathfrak{R}\left[ \left(p-q-\zeta\right) \frac{1+Az}{1+Bz}+\left(q+\zeta+\upsilon\right) \right] >0, $$

whenever (38) holds. Then \(f(z)\in F_{p,\upsilon }\left (\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \right) \ \)from Lemma 2. This proves (i). To prove (ii), we assume that (39) holds, then all the assumptions of Lemma 3 are satisfied for \(\widetilde {\beta },\ \overline {\gamma }\ \) which stated above and \(\widetilde {\alpha }=\frac {1-A}{1-B}.\ \)It follows that

$$\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;A,B\right) \subset\mathcal{S}_{p,q}^{\lambda,\mu,\eta}\left(\zeta;\tau\left(A,B\right)\right), $$

where τ(A,B) given by (40) is the best possible □

Conclusion

In our present investigation, we have derived some subordination results of certain subclasses of multivalent analytic functions which are defined by a generalized fractional differintegral operator. We have also successfully considered inclusion relations for functions in the class \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right)\) and the images of these functions by the generalized Bernardi-Libera-Livingston integral operator.

Abbreviations

\(\mathcal {A}(p)\) :

The class of analytic and multivalent functions in the open unit disc \(\mathbb{U}=\{z\in \mathbb{C}|z| <1\}\)

∗:

Convolution of two power series

≺:

Subordination of two analytic functions in \(\mathbb {U\ }\)

2 F 1(a,b;c;z)(c≠0,−1,−2,…):

The well-known (Gaussian) hypergeometric function

\(J_{0,z}^{\lambda,\mu,\eta,p}f(z)\) :

The generalized fractional derivative operator for \(f\in \mathcal {A}(p)\)

F p, υ :

The generalized Bernardi-Libera-Livingston integeral operator

References

  1. 1

    Bulboacă, T.: Differential subordinations and superordinations, recent results, House of Scientific Book Publ.House of Scientific Book Publ., Cluj-Napoca (2005).

    Google Scholar 

  2. 2

    Miller, S. S., Mocanu, P. T.: Differential subordination: theory and applications, series on monographs and textbooks in pure and applied mathematics, Vol. 225. Marcel Dekker Inc., New York and Basel (2000).

    Google Scholar 

  3. 3

    Whittaker, E. T., Watson, G. N.: A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; With an Account of the Principal Transcendental Functions, Gourth Edition. Cambridge Univ. Press, Cambridge (1927).

    Google Scholar 

  4. 4

    Aouf, M. K., Mostafa, A. O., Zayed, H. M.: Some characterizations of integral operators associated with certain classes of p-valent functions defined by the Srivastava-Saigo-Owa fractional differintegral operator. Complex Anal. Oper. Theory. 10(6), 1267–1275 (2016).

    MathSciNet  Article  Google Scholar 

  5. 5

    Aouf, M. K., Mostafa, A. O., Zayed, H. M.: Subordination and superordination properties of p-valent functions defined by a generalized fractional differintegral operator. Quaest. Math. 39(4), 545–560 (2016).

    MathSciNet  Article  Google Scholar 

  6. 6

    Aouf, M. K., Mostafa, A. O., Zayed, H. M.: On certain subclasses of multivalent functions defined by a generalized fractional differintegral operator. Afr. Mat. 28(1-2), 99–107 (2017).

    MathSciNet  Article  Google Scholar 

  7. 7

    Mostafa, A. O., Aouf, M. K., Zayed, H. M., Bulboacă, T.: Multivalent functions associated with Srivastava-Saigo-Owa fractional differintegral operator. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 112, 1409–1429 (2018).

    MathSciNet  Article  Google Scholar 

  8. 8

    Mostafa, A. O., Aouf, M. K., Zayed, H. M.: Inclusion relations for subclasses of multivalent functions defined by Srivastava-Saigo-Owa fractional differintegral operator. Afr. Mat. 29(3-4), 655–664 (2018).

    MathSciNet  Article  Google Scholar 

  9. 9

    Mostafa, A. O., Aouf, M. K., Zayed, H. M.: Subordinating results for p-valent functions associated with the Srivastava-Saigo-Owa fractional differintegral operator. Afr. Mat. 29(5-6), 809–821 (2018).

    MathSciNet  Article  Google Scholar 

  10. 10

    Zayed, H. M., Bulboacă, T.: Sandwich results for higher order fractional derivative operators. Mat. Stud. 49, 52–66 (2018).

    MathSciNet  MATH  Google Scholar 

  11. 11

    Zayed, H. M., Aouf, M. K., Mostafa, A. O.: Subordination properties for multivalent functions associated with a generalized fractional differintegral operator. Axioms. 7(no. 2) (2018). https://doi.org/10.3390/axioms7020027.

    Article  Google Scholar 

  12. 12

    Zayed, H. M., Mohammadein, S. A., Aouf, M. K.: Sandwich results of p-valent functions defined by a generalized fractional derivative operator with application to vortex motion. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 113, 1499–1514 (2019). https://doi.org/10.1007/s13398-018-0559-z.

    MathSciNet  Article  Google Scholar 

  13. 13

    Owa, S.: On the distortion theorems. I, Kyungpook. Math. J. 18, 53–59 (1978).

    MathSciNet  MATH  Google Scholar 

  14. 14

    Goyal, S. P., Prajapat, J. K.: A new class of analytic p−valent functions with negative coefficients and fractional calculus operators, Tamsui Oxford Univ. J. Math. Sci. 20(2), 175–186 (2004).

    MathSciNet  MATH  Google Scholar 

  15. 15

    Hallenbeck, D. Z., Ruscheweyh, St.: Subordination by convex functions. Proc. Amer. Math. Soc. 52, 191–195 (1975).

    MathSciNet  Article  Google Scholar 

  16. 16

    Eenigenburg, P., Miller, S. S., Mocanu, P. T., Reade, M. O.: On a Briot-Bouquet differential subordination. In: General Inequalities 3, ISNM, pp. 339–348. Birkhauser, Verlag, Basel (1983).

    Google Scholar 

  17. 17

    Mocanu, P. T., Ripeanu, D., Serb, I.: The order of stadikeness of certain integral operators. Math. (Cluj). 23(2), 225–230 (1981).

    MATH  Google Scholar 

  18. 18

    Choi, J. H., Saigo, M., Srivastava, H. M.: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 276(1), 432–445 (2002).

    MathSciNet  Article  Google Scholar 

  19. 19

    MacGregor, T. H.: Radius of univalence of certain analytic functions. Proc. Amer. Math. Soc. 14, 514–520 (1963).

    MathSciNet  Article  Google Scholar 

  20. 20

    Suffridge, T. J.: Some remarks on convex maps of the unit disk. Duke Math. J. 37, 775–777 (1970).

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author would like to express her sincere thanks to Hillal Elshehabey from Mathematics department, Faculty of Science, South Valley university, for his help in plotting the function h(z), as well as the referees for their valuable suggestions and comments, which enhanced the paper.

Funding

The author declares that she had no funding.

Author information

Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin Elkom, 32511, Egypt

    Hanaa M. Zayed

Authors
  1. Hanaa M. Zayed
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The author read and approved the final manuscript.

Corresponding author

Correspondence to Hanaa M. Zayed.

Ethics declarations

Competing interests

The author declares that she have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zayed, H.M. Subordination and inclusion theorems for higher order derivatives of a generalized fractional differintegral operator. J Egypt Math Soc 27, 17 (2019). https://doi.org/10.1186/s42787-019-0020-2

Download citation

Keywords

  • Differential subordination
  • Multivalent functions
  • Higher order derivatives
  • Generalized fractional differintegral operator

Mathematics Subject Classification

  • 30C45
  • 30C50.