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Differential subordination applications to a class of meromorphic multivalent functions associated with Mittag-Leffler function
Journal of the Egyptian Mathematical Society volume 27, Article number: 31 (2019)
Abstract
In this paper, using the principal of differential subordination, we obtain some properties of certain class of p-valent meromorphic functions, which are defined by Mittag-Leffler function.
Introduction
Denote by Σp,m the class of analytic meromorphic multivalent functions of the form:
where \(\mathbb {U}^{\ast }=\{z\in \mathbb {C} \) and \(0<|z|<1\}=\mathbb {U}\backslash \{0\}.\ \)We note that Sigmap,1−p=Σp.
For two functions f(z) and g(z), analytic in \(\mathbb {U}, f(z)\) is subordinate to g(z)(f(z)≺g(z)) in \(\mathbb {U}\), if there exists a function ω(z), analytic in \(\mathbb {U}\) with ω(0)=0 and \(\left \vert \omega (z)\right \vert <1, f(z)=g(\omega (z)) (z\in \mathbb {U})\) and if g(z) is univalent in \(\mathbb {U}\), then (see for details [1] and also [2])
The Hadamard product of f(z) and g(z) given by
is defined by
The Mittag-Leffler function Eα(z) (z\(\in \mathbb {U}^{\ast }\)) ([3] and [4]) see also ([5, 6] and [7]) is defined by
For \(\alpha,\beta,\gamma \in \mathbb {C}\), ℜ(α)>0, max { 0, ℜ(c)−1} and ℜ(c)>0, Srivastava and Tomovski [8] generalized Mittag-Leffler function by the function
and proved that it is an entire function in the complex z-plane, where
Mostafa and Aouf [9] (see also [10]) used the function \(E_{\alpha,\beta }^{\gamma,c}(z)\) and defined the meromorphic function
and for f(z)∈Σp,m, they defined the operator
From (5) it is easy to have
and
We note that:
(i) \(\mathcal {H}_{p,0,\beta }^{1,1}f(z)=f(z);\)
(ii) \(\mathcal {H}_{p,0,\beta }^{2,1}f(z)=\left (p+1\right) f(z)+zf^{^{\prime }}(z);\)
(iii) \(\mathcal {H}_{1,0,\beta }^{2,1}f(z)=2f(z)+zf^{^{\prime }}(z).\)
Using the operator \(\mathcal {H}_{p,\alpha,\beta }^{\gamma,c}f(z)\), we have the following definition.
Definition 1.
For fixed A and B (−1≤B<A≤1), we say that a function f∈Σp,m is in the class \(\Sigma _{p,m}^{\gamma,c}\left (\alpha,\beta ;A,B\right)\) if it satisfies
In view of the definition of differential subordination, (8) is equivalent to
We note that:
(i)
the class Σp(A,B) was introduced and studied by Mogra [11].
(ii)
Preliminary results
The following lemmas will be required in our investigation.
Lemma 1
[12]. Let h be a convex (univalent) function in \(\mathbb {U}\) with h(0)=1. Also let
be analytic in \(\mathbb {U}.\) If
then
Lemma 2
[13]. Let μbe a positive measure on the unit interval [0,1]. Let g(z,t)be a complex valued function defined on \(\mathbb {U}\times \left [ 0,1\right ] \) such that g(.,t) is analytic in \(\mathbb {U}\) for each t∈[0,1] and such that g(z,.) is μ integrable on [0,1] for all \(z\in \mathbb {U}.\) In addition, suppose that ℜ{g(z,t)}>0,g(−r,t)is real and
If the function G is defined by
then
Each of the identities (asserted by Lemma 2) is fairly well known (cf., e.g., [[8], ch. 14]).
Lemma 3
[14]. For real or complex numbers a, b, and c (c≠0,−1,−2,…)
and
Lemma 4
[15]. Let Φbe analytic in \(\mathbb {U}\) with
Then, for any function F, analytic in \(\mathbb {U}, \left (\Phi \ast F\right) \left (\mathbb {U}\right) \) is contained in the convex hull of \( F\left (\mathbb {U}\right).\)
Main inclusion relationships
Unless otherwise mentioned, we assume throughout this paper that \(-1\leq B<A\leq 1,\alpha,\beta,\gamma \in \mathbb {C}, \Re (\alpha)>0\), max { 0, ℜ(c)−1}, ℜ(c)>0,δ>0,f(z) given by (1) and \(z\in \mathbb {U}^{\ast }.\)
Theorem 1
Let γ≠0 and f(z) satisfy:
then
where
is the best dominant of (17). Furthermore,
where
The result is the best possible.
Proof
Let
where ϕ is given by (11). Differentiating (21) and using (6), we get
Now, by using Lemma 1 for \(\tau =\frac {\gamma }{\delta c},\) we get
This proves (17) of Theorem 1. In order to prove (20), we need to show that
We have
Putting
which is a positive measure on [0,1], we obtain
Then
Assuming r→1− in the above inequality, we obtain (22). The result in (19) is the best possible and Ψ is the best dominant of (17). This completes the proof of Theorem 1. □
Theorem 2
Let \(f(z) \in \Sigma _{p,m}^{\gamma,c}\left (\alpha,\beta,\eta \right) \left (0\leq \eta < p\right),\) then
where
The result is the best possible.
Proof
Since \(f(z)\in \Sigma _{p,m}^{\gamma,c}\left (\alpha,\beta,\eta \right),\) let
where u(z) in the form (11) and ℜ{u(z)}>0. Differentiating (25) and using (6), we get
Applying the following estimate [20],
in (26), we get
It is easily seen that the right-hand side of (27) is positive, if r<R, where R is given by (24). In order to show that the bound R is the best possible, we consider the function f∈Σp,m defined by
Noting that
for
This completes the proof of Theorem 2. □
Putting δ=1 in Theorem 2, we obtain the following result.
Corollary 1
If \(f(z) \in \Sigma _{p,m}^{\gamma,c}\left (\alpha,\beta,\eta \right) \left (0\leq \eta < p\right),\) then \(f(z) \in \Sigma _{p,m}^{\gamma +1,c}\left (\alpha,\beta,\eta \right) \) for |z|<R∗, where
The result is the best possible.
Theorem 3
If the function f(z)∈Σp,m satisfies
then
and
where Ψ1(z) is in the form (18) and ρ given by (20). The result is the best possible.
Proof
The proof follows by taking the same lines as in the proof of Theorem 1 and taking \(\phi (z)=z^{p}\mathcal {H}_{p,\alpha,\beta }^{\gamma,c}f(z)\) in (21). □
For the function f(z) in the class Σp,m, Kumar and Shukla [21] defined the integral operator гμ,p:Σp,m→Σp,m as follows:
From (28), we get
Theorem 4
Let the function f(z)given by (1) be in the class \( \Sigma _{p,m}^{\gamma,c}\left (\alpha,\beta ;A,B\right) \) and гμ,p(f)(z) defined by (28). Then
where
is the best dominant of (31). Furthermore,
where
The result is the best possible.
Proof
Let
where L in the form (11). Differentiating (34) and using (29), we get
Now the remaining part of Theorem 4 follows by using the technique used in proving Theorem 1. □
Theorem 5
Let the function гμ,p(f)(z) defined by (28) satisfy:
then
where
The result is the best possible.
Proof
Let
where K in the form (11). Differentiating (37) and using (29) and (35), we get
Now the remaining part of Theorem 5 follows by using the technique used in proving Theorem 1. □
Theorem 6
Let the function f(z)∈Σp,m satisfy:
then
where гμ,p(f)(z) is given by (28) and θ is given as in Theorem 5. The result is the best possible.
Proof
The proof follows by taking the same lines as in Theorem 5. □
Theorem 7
Let f(z)be in the class Σp,m. Also, let g(z)∈Σp,m satisfy:
If
then
where
Proof
Let
we note that ϕ is analytic in \(\mathbb {U}\), with ϕ(0)=0 and |ϕ(z)|≤|z|p+m. Then, by applying the familiar Schwarz Lemma [22], we have ϕ(z)=zp+mΨ(z) is analytic in \(\mathbb {U}\) and \(\left \vert \Psi (z)\right \vert \leq 1 \left (z\in \mathbb {U}\right).\) Therefore, (40) leads to
Differentiating (41), we obtain
Letting \(\chi (z)=z^{p}\mathcal {H}_{p,\alpha,\beta }^{\gamma,c}g(z),\) we see that the function χ is of the form (11) and is analytic in \( \mathbb {U}\), ℜ{χ(z)}>0 and
so, we find from (42) that
Using the following known estimates [23] (see also [20]),
in (43), we have
which is certainly positive, provided that r<R0,R0 given by (39). □
Theorem 8
Let the function f(z)∈Σp,m satisfy:
then
where ε in the form (20). The result is the best possible.
Proof
Let
where ϕ in the form (11). Differentiating (44) and using (6), we have
Now the remaining part of Theorem 8 follows by using the technique used in proving Theorem 1, and using the inequality:
we have the result asserted by Theorem 8. □
Theorem 9
Let the function \(f(z)\in \Sigma _{p,m}^{\gamma,c}\left (\alpha, \beta ;A,B\right)\) and let g(z)∈Σp,m satisfy:
Then,
Proof
We have
Since
and \(\frac {1+Az}{1+Bz}\) is convex in \(\mathbb {U}\), it follows from (8) and Lemma 4 that \(\left (f\ast g\right) (z)\in \Sigma _{p,m}^{\gamma,c}\left (\alpha,\beta ;A,B\right),\) which completes the proof of Theorem 9. □
Remark 1
For different value of γ, c, α, β, and p in the above results, we obtain results corresponding to the functions given in the introduction.
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Mostafa, A.O., El-hawsh, G.M. Differential subordination applications to a class of meromorphic multivalent functions associated with Mittag-Leffler function. J Egypt Math Soc 27, 31 (2019). https://doi.org/10.1186/s42787-019-0028-7
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DOI: https://doi.org/10.1186/s42787-019-0028-7
Keywords
- Analytic function
- Convex function
- Meromorphic multivalent functions
- Subordination and Mittag-Leffler function