Some inequalities on Bazilevič class of functions involving quasi-subordination

K. R. Karthikeyan1, G. Murugusundaramoorthy2 and N. E. Cho3,∗ 1 Department of Applied Mathematics and Science, National University of Science & Technology (By Merger of Caledonian College of Engineering and Oman Medical College), Sultanate of Oman 2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology,Deemed to be University, Vellore, Tamilnadu, India 3 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea


Introduction
Bazilevič [3] introduced the class B(α, ζ, g) of functions which is defined by the integral where p ∈ P, the class of analytic function with positive real part and g ∈ S * , the well-known class of starlike function. The numbers α > 0 and ζ are real and all powers are chosen so that it remains single-valued. Apart from the fact that B(α, ζ, g) is univalent, we have little or no information on these family of functions. But if we simplify, for example, letting ζ = 0 and g(z) = z we get the well-known class B(α) which is given by where f ∈ A is the class of functions having a Taylor series expansion of the form f (z) = z + ∞ k=2 a k z k (z ∈ U = {z : |z| < 1}) . (1.1) Let 0 ≤ η < 1, S * (η) and C(η) symbolize the classes of starlike functions of order η and convex functions of order η, respectively. Let S * (η, ϑ) (see [9]) denote the class of functions f ∈ A satisfying the inequality Robertson [18] introduced quasi-subordination unifying the concept of subordination and majorization.
For analytic functions f and g in U, f is quasi-subordinate to g in U, denoted by f ≺ q g, if there exist a Schwarz function w and an analytic function φ satisfying |φ(z)| < 1 and f (z) = φ(z)g(w(z)) in U. If φ(z) = 1, quasi-subordination reduces to subordination. If we let w(z) = z, then quasi-subordination reduces to the concept of majorization. For f ∈ A given by (1.1) and 0 < q < 1, the Jackson's q-derivative operator or q-difference operator is defined by (see [1,2] ) and note that lim q→1 − D q f (z) = f (z). Notations and symbols play an very important role in the study of q-calculus. Throughout this paper, we let ( Let q-analogue incomplete beta function χ(z) (see [19]) is defined by . Lately, the study of the q − calculus has riveted the rigorous consecration of researchers. The great attention is because of its gains in many areas of mathematics and physics. The significance of the q − derivative operator D q is quite evident by its applications in the study of several subclasses of analytic functions. Initially, in the year 1990, Ismail et al. [5] gave the idea of q − starlike functions. Nevertheless, a firm base of the usage of the q − calculus in the context of Geometric Function Theory was efficiently established, and the use of the generalized basic (or q−) hypergeometric functions in Geometric Function Theory was made by Srivastava (see, for details, [23]).The study of geometric function theory in dual with quantum calculus was initiated by Srivastava ( [24], also see [25]). After that, extraordinary studies have been done by many mathematicians, which offer a significant part in the encroachment of Geometric Function Theory. In particular, Srivastava et al. [26][27][28][29][30][31][32] also considered some function classes of q − starlike functions related with conic region and focussed upon the classes of q − starlike functions related with the Janowski functions from several different aspects. Inspired by aforementioned works on q − calculus we now define the q-analogue of the function which maps U onto a conic region. Let (1.6) The function ψ defined by (1.6) is the q-analogue of h(z) = z + 3 √ 1 + z 3 which maps the unit disc onto a leaf-like shaped region which is analytic and univalent. For details of functions mapping unit disc onto a leaf-like domain, refer to [20].
For functions f ∈ A given by (1.1) and h ∈ A of the form the Hadamard product (or convolution) is defined by We now introduce the following class of functions.
and H = f * h defined as in (1.8), let B t λ (γ; ψ) be the class of functions defined by where ψ ∈ P and has a series expansion of the form (1.10) Remark 1.1. Several well-known classes can be seen as special case of B t λ (γ; ψ) (see [7,15,17,22]). Here we highlight only the recent works which are associated with a conic region.
where ML c b (t; ψ) is the class recently introduced and studied by Murugusundaramoorthy and Bulboacȃ [12].
where S is the class of all univalent functions in A. The class R(ψ) was recently introduced by Khan et al. [8]. Further, we note that 1 + z 3 , the class of functions recently studied by Priya and Sharma [13].
] of all functions satisfying the above subordination condition was introduced and studied by Malik et al. [10].

Prelimnaries
In this section, we state the results that would be used to establish our main results which can be found in the standard text on univalent function theory.
Lemma 2.1. [4] If the function f ∈ A given by (1.1) and g(w) given by (2.2) Remark 2.1. The elements of the determinant in (2.2) are given by Lemma 2.2. [14] If p(z) = 1 + ∞ k=1 p k z k ∈ P, then |p k | ≤ 2 for all k ≥ 1, and the inequality is sharp for p(z) = p 1 (z) = 1+z 1−z . Lemma 2.3. [11] Let p ∈ P and also let v be a complex number. Then The result is sharp for functions given by

Coefficients estimates for functions in
Hereafter, unless otherwise mentioned we assume that The class of all functions in B t λ (γ; ψ) is not univalent, so the inverse is not guaranteed. However, there exist an inverse function in some small disk with center at w = 0 depending on the parameters involved.
, then the estimates of the inverse coefficients of f are and Proof. Let f ∈ B t λ (γ; ψ). Then by the definition of quasi-subordination, there is a function w(z) such that Define the function p by We can note that p(0) = 1 and p ∈ P (see Lemma 2.2). Using (3.5), it is easy to see that So we have (3.6) The left hand side of (3.6) will be where Γ k 's are the corresponding coefficients from the power series expansion of h, which may be real or complex. By using (3.6) and (3.7), we have This completes the proof of the Theorem 3.1.

Theorem 3.2.
If the function f given by (1.1) and g given by (2.1) are inverse functions and if f ∈ A satisfies the inequality (3.10) then the estimates of the inverse coefficients of f satisfying the inequality (3.10) are Proof. From the equivalent subordination condition proved by Kuroki and Owa in [9], we may rewrite the conditions (3.10) in the form (3.11) Further, we note that maps U onto a convex domain conformally and is of the form Substituting the values of A 1 , A 2 , d 0 = 1 and d 1 = 0 in Theorem 3.1, we have the assertion of the theorem.
If we let h(z) = z + ∞ n=2 z n , t = λ = 0 and q → 1 − in Theorem 3.2, we get the following result obtained by Sim and Kwon [21].
If we let t = 0 and λ = 0 in Theorem 3.4, we have the following result obtained by Ramachandran et al. [16].
Remark 3.1. Some subordination results for the well-known Janowski class with the function κ defined by was recently studied by Malik et al. [10].
Theorem 3.6. Suppose that f ∈ B t λ (γ; ψ) with ψ(z) of the form where −1 ≤ B < A ≤ 1 and κ(z) is defined as in (3.13), then the estimates of the inverse coefficients of f are Proof. Following the steps as in Theorem 1 of [8], we get (3.14) Now replacing A 1 , A 2 and A 3 in Theorem 3.1 with the corresponding coefficients of the series given in (3.14), we have the assertion of the theorem.
Corollary 3.8. [8] Suppose that f ∈ A satisfies the condition Then .
Remark 3.2. If we let q → 1 − in Corollary 3.8, we get the corresponding result of Priya and Sharma [13].

Fekete-Szegö problem for functions in
The Fekete-Szegö problem which is related to the Bieberbach conjecture represents various geometric quantities. The motivation to provide a unified approach to the Fekete-Szegö problem and initial coefficients was from the study due to Kanas [6]. Note that Theorem 4.1 is a generalization of result obtained in [6].
where ρ is given by 2) The inequalities are sharp for each µ.
[33] Suppose f (z) = z + a 2 z 2 + a 3 z 3 + · · · ∈ S * (ψ) (z ∈ U). Then The inequality is sharp for the function given by Conclusion 4.1. By defining Bazilevič functions of complex order using quasi-subordination and Hadamard product, we were able to unify and extend various classes of analytic function. New extensions were discussed in detail. Further, by replacing the ordinary differentiation with quantum differentiation we have attempted at the discretization of some of the well-known results.