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Fekete-Szegő inequalities for certain class of analytic functions connected with q-analogue of Bessel function
Journal of the Egyptian Mathematical Society volume 27, Article number: 42 (2019)
Abstract
In this paper, we obtain Fekete-Szegő inequalities for a certain class of analytic functions f satisfying \(1+\frac {1}{\zeta }\left [\frac {z\left (\mathcal {N}_{\nu,q}^{\lambda }f(z)\right)^{\prime }} {(1-\gamma)\mathcal {N} _{\nu,q}^{\lambda }f(z)+\gamma z\left (\mathcal {N}_{\nu,q}^{\lambda }f(z) \right)^{\prime }}-1\right ]\prec \Psi (z)\). Application of our results to certain functions defined by convolution products with a normalized analytic function is given, and in particular, Fekete-Szegő inequalities for certain subclasses of functions defined through Poisson distribution are obtained.
Introduction
Let \(\mathcal {A}\) denote the class of analytic functions of the form:
and \(\mathcal {S}\) be the subclass of \(\mathcal {A}\) which are univalent functions in \(\mathbb {D}\).
If \(k\in \mathcal {A}\) is given by:
then, the Hadamard (or convolution) product of f and k is defined by:
If f and F are analytic functions in \(\mathbb {D}\), we say that fis subordinate toF, written f≺F, if there exists a Schwarz functionw, which is analytic in \(\mathbb {D}\), with w(0)=0, and |w(z)|<1 for all \(z\in \mathbb {D}\), such that \(f(z)=F(w(z)), z\in \mathbb {D}\). Furthermore, if the function F is univalent in \(\mathbb {D} \), then we have the following equivalence (see [1] and [2]):
The Bessel function of the first kind of order ν is defined by the infinite series:
where Γ stands for the Gamma function. Recently, Szász and Kupán [3] investigated the univalence of the normalized Bessel function of the first kind \(g_{\nu }:\mathbb {D}\rightarrow \mathbb {C}\) defined by (see also [4–6])
For 0<q<1, the q-derivative operator for gν is defined by:
where
Using definition formula (4), we will define the next two products:
(i) For any non-negative integer k, the q-shifted factorial is given by:
(ii) For any positive number r, the q-generalized Pochhammer symbol is defined by:
For ν>0,λ>−1, and 0<q<1, define the function \(\mathcal {I} _{\nu,q}^{\lambda }:\mathbb {D}\rightarrow \mathbb {C}\) by:
Remark 1
A simple computation shows that:
where the function \(\mathcal {M}_{q,\lambda +1}\) is given by:
Using the definition of q-derivative along with the idea of convolutions, we introduce the linear operator \(\mathcal {N}_{\nu,q}^{\lambda }:\mathcal {A} \rightarrow \mathcal {A}\) defined by:
where
Remark 2
From definition relation (5), we can easily verify that the next relations hold for all \(f\in \mathcal {A}\):
(i) \([\lambda +1,q]\mathcal {N}_{\nu,q}^{\lambda }f(z)=[\lambda,q]\mathcal {N} _{\nu,q}^{\lambda +1}f(z) +q^{\lambda }z\partial _{q}\left (\mathcal {N} _{\nu,q}^{\lambda +1}f(z)\right), z\in \mathbb {D}\);
(ii) \(\lim \limits _{q\to 1^{-}}\mathcal {N}_{\nu,q}^{\lambda }f(z)= \mathcal {I} _{\nu,1}^{\lambda }\times f(z)=:\mathcal {I}_{\nu }^{\lambda }f(z)=\)
\(z+\sum \limits _{k=2}^{\infty }\frac {k!}{(\lambda +1)_{k-1}}\frac {(-1)^{k-1} \Gamma (\nu +1)}{4^{k-1}(k-1)!\Gamma (k+\nu)}\,a_{k}z^{k}, z\in \mathbb {D}\).
Now, we define the class of functions \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\) as follows:
Definition 1
Let \(\Psi (z):=1+B_{1}z+B_{2}z^{2}+\dots, z\in \mathbb {D }\), with B1>0, be a starlike (univalent) function with respect to 1, which maps the unit disk \(\mathbb {D}\) onto a region included in the right half plane which is symmetric with respect to the real axis. For \(\zeta \in \mathbb {C}^{\ast }\), and 0≤γ<1, the function \(f\in \mathcal {A}\) is said to be in the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\)if the function
is analytic in \(\mathbb {D}\) and satisfies:
Putting q→1−, we obtain that \(\lim \limits _{q\to 1^{-}}\mathcal {M} _{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)=:\mathcal {G}_{\nu }^{\lambda,\gamma }(\zeta ;\Psi)\), where
In this paper, we obtain the Fekete-Szegő inequalities for the functions of the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\). We give some application of our results to certain functions defined by convolution products with a normalized analytic function. In particular, Fekete-Szegő inequalities for certain subclasses of functions defined through Poisson distribution are obtained.
Fekete-Szegő problem
Denoted by \(\mathcal {P}\), the well-known Carathéodory’s class of analytic functions in \(\mathbb {D}\), normalized with P(0)=1, and having positive real part in \(\mathbb {D}\), that is ReP(z)>0 for all \(z\in \mathbb {D}\) (see [7]).
To prove our results, we need the following two lemmas.
Lemma 1
[8, Lemma 3] If \(p(z)=1+c_{1}z+c_{2}z^{2}+\dots \in \mathcal {P}\), and α is a complex number, then
Lemma 2
[9, Lemma 1] If \(p(z)=1+c_{1}z+c_{2}z^{2}+\dots \in \mathcal {P}\), then
When α<0 or α>1, the equality holds if and only if \(p(z)= \frac {1+z}{1-z}\) or one of its rotations.
If 0<α<1, then the equality holds if and only if \(p(z)=\frac {1+z^{2} }{1-z^{2}}\) or one of its rotations.
If α=0, the equality holds if and only if:
or one of its rotations.
If α=1, the equality holds if and only if:
Like it was mentioned in [9, pages 162–163], although the above upper bound is sharp, it can be improved as follows when 0<α<1:
and
Theorem 1
If the function f given by (1) belongs to the class \( \mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1, and μ is a complex number, then:
where ψk,k∈{2,3}, are given by (6).
Proof
If \(f\in \mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\), then there exists a Schwarz function w, that is w is analytic in \(\mathbb {D}\), with w(0)=0 and \(\left |w(z)\right |<1, z\in \mathbb {D}\), such that:
Since w is a Schwarz function, it follows that the function p1 defined by:
belongs to \(\mathcal {P}\). Defining the function p by:
in view of (9) and (10), we have:
From (10), we easily get:
therefore,
and from (12), we obtain:
On the other hand, from (11), according to (5), it follows that
and combining (13) with (14), we have:
and
Therefore,
where
and from Lemma 1, our result follows immediately. □
Putting q→1− in Theorem 1, we obtain the next corollary:
Corollary 1
If the function f given by (1) belongs to the class \(\mathcal {G}_{\nu }^{\lambda,\gamma }(\zeta ;\Psi)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1, and μ is a complex number, then
Using a similar proof like for Theorem 1 combined with Lemma 2, we can obtain the following theorem:
Theorem 2
If the function f given by (1) belongs to the class \( \mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1 and \(\mu,B_{2}\in \mathbb {R}\), and ζ>0, then
with
and
where ψk,k∈{2,3}, are given by (6).
Proof
With the same proof like those of Theorem 1, we obtain the equalities (16) and (17) hold.
(i) According to the first part of Lemma 2, we have:
Using (17), simple computation shows that the inequality α≤0 is equivalent to μ≤σ1, and from (16) combined with the inequality \(\left |c_{2}-\alpha c_{1}^{2}\right |\leq -4\alpha +2\), the first of our theorem is proved.
(ii) The second part of Lemma 2 shows that:
From (17), it is easy to check that the inequality 0≤α≤1 is equivalent to σ1≤μ≤σ2. From the relation (16), the inequality \(\left |c_{2}-\alpha c_{1}^{2}\right |\leq 2\) proves the second part of our result.
(iii) Finally, form the third part of Lemma 2, we have:
The relation (17) shows immediately that α≥1 is equivalent to μ≥σ2, while (16) combined with the inequality \(\left |c_{2}-\alpha c_{1}^{2}\right |\leq 4\alpha -2\) proves the last part of our result. □
Taking q→1− in Theorem 2, we get the next special case:
Corollary 2
If the function f given by (1) belongs to the class \(\mathcal {G}_{\nu }^{\lambda,\gamma }(\zeta ;\Psi)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1 and \(\mu,B_{2}\in \mathbb {R}\), and ζ>0, then
with
and
With a similar proof like for Theorem 1 and using the inequalities (7) and (8), we obtained the next result.
Theorem 3
If the function f given by (1) belongs to the class \( \mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1 and \(\mu,B_{2}\in \mathbb {R}\), and ζ>0, then the next inequalities hold:
(i) for σ1<μ≤σ3, we have
(ii) for σ3≤μ≤σ2, we have
where σ1 and σ2 are defined by (18) and (19), respectively,
and ψk,k∈{2,3}, are given by (6).
Proof
With the same computations like in the proof of Theorem 1, we obtain the relations (16) and (17), while (15) is equivalent to:
(i) To prove the first part of our theorem, we will use the inequality (7). Thus, according to (16), (17), and the above relation, it is easy to check that (7) could be written in the equivalent form (22), while the assumption \(0<\alpha \leq \frac {1 }{2}\) is equivalent to σ1<μ≤σ3.
(ii) For the proof of the second part of our result, we will use the inequality (8). From (16), (17), and (24), it follows that (8) could be written in the form (23), and the assumption \(\frac {1}{2}\leq \alpha <1\) is equivalent to σ3<μ≤σ2. □
Putting q→1− in Theorem 3, we obtain the following result:
Corollary 3
If the function f given by (1) belongs to the class \(\mathcal {G}_{\nu }^{\lambda,\gamma }(\zeta ;\Psi)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1 and \(\mu,B_{2}\in \mathbb {R}\), and ζ>0, then the next inequalities hold:
(i) for η1<μ≤η3, we have
(ii) for η3≤μ≤η2, we have
where η1 and η2 are defined by (20) and (21), respectively, and
Applications to functions defined by poisson distribution
In [10], Porwal studied a power series whose coefficients are probabilities of the Poisson distribution, that is:
and motivated by this investigation Srivastava and Porwal [11] introduced the linear operator \(\mathcal {I}^{m}:\mathcal {A}\rightarrow \mathcal {A}\) defined by:
where \(f\in \mathcal {A}\) has the form (1).
Definition 2
Let the function Ψ satisfying the conditions of Definition 1. For \(\zeta \in \mathbb {C}^{\ast }, 0\leq \gamma <1\), and \(k\in \mathcal {A}\), the function \(f\in \mathcal {A}\) is said to be in the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;k; \Psi \right)\) if \(f\times k\in \mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\), that is"
is analytic in \(\mathbb {D}\) and satisfies
A special case of the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;k;\Psi \right)\) is obtained for k=Im; hence, \(f\in \mathcal { M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;\mathrm {I}_{m};\Psi \right)\) if and only if \(\mathcal {I}^{m}f\in \mathcal {M}_{\nu,q}^{\lambda,\gamma }(\zeta ;\Psi)\).
Applying Theorems 1 and 2 for the function f×k given by (3), we get the following results, respectively:
Theorem 4
If the function f given by (1) belongs to the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;k;\Psi \right)\), with \( \Psi (z)=1+B_{1}z+B_{2}z^{2}+\dots, k\in \mathcal {A}\) is given by (2) with b2b3≠0, and μ is a complex number, then
where ψk and k∈{2,3} are given by (6).
Theorem 5
If the function f given by (1) belongs to the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;k;\Psi \right)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1 and \(\mu,B_{2}\in \mathbb {R}, k\in \mathcal {A}\) is given by (2) with b2b3≠0, and ζ>0, then
with
and
and ψk,k∈{2,3}, are given by (6).
For k:=Im, we have
and for this special case from Theorems 4 and 5, we deduce to the following results, respectively:
Theorem 6
If the function f given by (1) belongs to the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;\mathrm {I}_{m};\Psi \right)\), with Ψ(z)=1+B1z+B2z2+…, and μ is a complex number, then
where ψk and k∈{2,3} are given by (6).
Theorem 7
If the function f given by (1) belongs to the class \(\mathcal {M}_{\nu,q}^{\lambda,\gamma }\left (\zeta ;\mathrm {I}_{m};\Psi \right)\), with Ψ(z)=1+B1z+B2z2+… satisfying the conditions of Definition 1 and \(\mu,B_{2}\in \mathbb {R}\), and ζ>0, then
with
and
where ψk and k∈{2,3} are given by (6).
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References
Bulboacă, T.: Differential subordinations and superordinations. Recent Results, House of Scientific Book Publ., Cluj-Napoca (2005).
Miller, S. S., Mocanu, P. T.: Differential subordinations: theory and applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225. Marcel Dekker Inc., New York and Basel (2000).
Szász, R., Kupán, P. A.: About the univalence of the Bessel functions, Studia Univ. Babeş-Bolyai Math. 54(1), 127–132 (2009).
Baricz, Á.: Geometric propertis of generalized Bessel functions. Publ. Math. Debr. 73, 155–178 (2008).
Jackson, F. H.: The application of basic numbers to Bessel’s and Legendre’s functions. Proc. Lond. Math. Soc. 3(2), 1–23 (1905).
Selvakumaran, K. A., Szász, R.: Certain geometric properties of an integral operator involving Bessel functions. Kyungpook Math. J. 58, 507–517 (2018).
Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907).
Libera, R. J., Zlotkiewicz E.J.: Coefficient bounds for the inverse of a function with derivative in \(\mathcal {P}\). Proc. Amer. Math. Soc. 87(2), 251–257 (1983).
Ma, W., Minda, D.: A unified treatment of some special classes of univalent functions. In: Li, Z., Ren, F., Lang, L., Zhang, S. (eds.)Proceedings of the Conference on Complex Analysis, Tianjin, 1992, pp. 157–169. Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge (1994).
Porwal, S.: An application of a Poisson distribution series on certain analytic functions. J. Complex Anal., 1–3 (2014). Art. ID 984135. https://doi.org/10.1155/2014/984135.
Srivastava, D., Porwal, S.: Some sufficient conditions for Poisson distribution series associated with conic regions. Int. J. Adv. Technol. Eng. Sci. 3(1), 229–235 (2015).
Acknowledgements
The authors are grateful to the reviewer of this article, that gave valuable remarks, comments, and advices, in order to revise and improve the results of the paper.
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Faculty of Science, Damietta University, New Damietta, Egypt.
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M. El-Deeb, S., Bulboacă, T. Fekete-Szegő inequalities for certain class of analytic functions connected with q-analogue of Bessel function. J Egypt Math Soc 27, 42 (2019). https://doi.org/10.1186/s42787-019-0049-2
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DOI: https://doi.org/10.1186/s42787-019-0049-2
Keywords
- Fekete-Szegő inequality
- Differential subordination
- Bessel function of first kind
- q-derivative
- Poisson distribution