Fekete-Szego\u030b inequalities for certain class of analytic functions connected with q-analogue of Bessel function

Sheza M El-Deeb,Teodor Bulboacă

doi:10.1186/s42787-019-0049-2

Abstract

In this paper, we obtain Fekete-Szegő inequalities for a certain class of analytic functions f satisfying 1+frac {1}{zeta }left [frac {zleft (mathcal {N}_{nu,q}^{lambda }f(z)right)^{prime }} {(1-gamma)mathcal {N} _{nu,q}^{lambda }f(z)+gamma zleft (mathcal {N}_{nu,q}^{lambda }f(z) right)^{prime }}-1right ]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.

Highlights

Let A denote the class of analytic functions of the form: ∞f (z) = z + akzk, z ∈ D := {z ∈ C : |z| < 1}, (1)k=2 and S be the subclass of A which are univalent functions in D
If f and F are analytic functions in D, we say that f is subordinate to F, written f ≺ F, if there exists a Schwarz function w, which is analytic in D, with w(0) = 0, and |w(z)| < 1 for all z ∈ D, such that f (z) = F(w(z)), z ∈ D
Szász and Kupán [3] investigated the univalence of the normalized Bessel function of the first kind gν : D → C defined by

Summary

Introduction

Let A denote the class of analytic functions of the form:. k=2 and S be the subclass of A which are univalent functions in D. Szász and Kupán [3] investigated the univalence of the normalized Bessel function of the first kind gν : D → C defined by (see [4,5,6]). Using definition formula (4), we will define the two products: (i) For any non-negative integer k, the q-shifted factorial is given by:. Using the definition of q-derivative along with the idea of convolutions, we introduce the linear operator Nνλ,q : A → A defined by: Nνλ,qf (z) := Iνλ,q(z) × f (z) = z + ψkakzk, z ∈ D,. Z ∈ D; we define the class of functions Mλν,,qγ (ζ ; ) as follows: Definition 1 Let (z) := 1 + B1z + B2z2 + . Ν > 0, λ > −1, 0 < q < 1, ζ ∈ C∗, 0 ≤ γ < 1

Putting q

If α