CONVEXITY OF INTEGRAL OPERATORS INVOLVING DINI FUNCTIONS

In this article, we are mainly interested to find some covexity properties for certain families of integral operators involving Dini functions in the open unit disc. The main tool in the proofs of our results are some functional inequalities of Dini functions. Some particular cases involving Dini functions are also a part of our investigations.


INTRODUCTION
Let A denote the class of functions f of the form which are analytic in the open unit disc U = {z : |z| < 1} and S denote the class of all functions which are univalent in U. Let S * (α) , C (α) and K (α) denote the classes of starlike, convex and close-to-convex functions of order α and are defined as: and It is clear that S * (0) = S * , C (0) = C and K (0) = K.
Special functions have great importance in pure and applied mathematics.The widely use of these functions have attracted many researchers to work on the different directions.Geometric properties of special functions such as Hypergeometric functions, Bessel functions, Struve functions, Mittage-Lefller functions, Wright functions and some other related functions is an ongoing part of research in geometric function theory.We refer for some geometric properties of these functions [1,2,5,6,11] and references therein.
The Bessel function of the first kind J v is defined by where Γ stands for Euler gamma function.It is a particular solution of the second order linear homogeneous differential equation where v ∈ C. For some details see [2,13].Bessel functions are indispensable in many branches of pure and applied mathematics.Thus, it is important to study their properties in many aspects.Recently Baricz et al [3] studied the close-to-convexity of Dini functions and some monotonicity properties and functional inequalities for the modified Dini function are discussed in [4].Further some geometric properties of Dini functions are studied in [3,4,7].Now, we consider the normalized Dini functions q υ : U → C defined as (−1) n (2n + 1) Γ (υ + 1) 3) The Pochhammer (or Appell) symbol, defined in terms of Euler's gamma functions is given as (x) n = Γ(x + n)/Γ(x) = x(x + 1)...(x + n − 1).
Recently, Deniz et al. [8], Din et al. [9], Din et al. [10] and Srivastava et al. [12] have obtained sufficient conditions for the univalence of certain families of integral operators defined by Bessel, Dini, Struve and Mittage-Leffler functions respectively.The families of integral operators are defined below: and In this paper, we are mainly interested in the convexity of the integral operators involving Dini function q v .These integral operators are defined as and Now we prove some functional inequalities which are useful in establishing our main results.
Lemma 1.1.Let v ∈ R and consider the normalized Dini function q υ : U → C, defined by where J υ is the Bessel function of first kind.Then the following inequalities hold for all z ∈ U. (i) Proof.(i) By using the well known triangle inequality with the equality and the inequality Furthermore, if we use the reverse triangle inequality and the inequality By combining the above inequalities, we get .
(ii) By using the well known triangle inequality with equality and the inequality we have Furthermore, if we use the reverse triangle inequality and the inequality then we get 2 By combining the above inequalities, it can be easily obtained

Convexity of Integral Operators Defined by Generalized Dini Functions
The main objective of this paper is to give convexity properties of integral operators involving Dini function.The main results are given as follows.