Certain Geometric Properties of Generalized Dini Functions

Muhey U. Din, Mohsan Raza , Saqib Hussain , andMaslina Darus 3 1Department of Mathematics, Government College University, Faisalabad, Pakistan 2Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan 3Faculty of Science and Technology, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Correspondence should be addressed to Mohsan Raza; mohsan976@yahoo.com


Introduction
Let A be the class of functions  of the form ( It is clear that S * (1) = S * (0) = S * , If  and  are analytic functions, then the function  is said to be subordinate to , written as () ≺ (), if there exist a Schwarz function  which is analytic with (0) = 0 and || < 1 such that () = (()).Furthermore, if 2 Journal of Function Spaces the function  is univalent in U then we have the following equivalent relation:  () ≺  () ⇐⇒  (0) =  (0) ,  (U) ⊂  (U) .
(5) Special functions have great importance in pure and applied mathematics.The wide use of these functions has attracted many researchers to work on the different directions.Geometric properties of special functions such as hypergeometric functions, Bessel functions, Struve functions, Mittag-Leffler functions, Wright functions, and some other related functions are an ongoing part of research in geometric function theory.We refer to some geometric properties of these functions [1][2][3][4][5][6][7][8][9][10] and references therein.
Consider the second-order homogeneous differential equation  2   () +   () + ( 2 − V 2 + (1 − ) V)  () = 0, (6) where , , and V are complex numbers.The particular solution of the homogenous differential equation ( 6) is called the generalized Bessel functions of the first kind of order V.It is defined as where Γ(.) denotes the gamma function.The function  V,, unifies the Bessel, modified Bessel, and spherical Bessel functions.

Special cases
(1) For  =  = 1, we have the Bessel functions of first kind of order V, defined as (2) For  = 1,  = −1, we obtain the modified Bessel functions of first kind of order V whose series form is given as (3) For  = 2,  = 1, we have the spherical Bessel functions of first kind of order V, given as For more details about these functions, see [5,11].Recently, Deniz et al. [12] studied the function  V,, which is defined by the relation as where , , V ∈ C. Using the well-known Pochhammer symbol we obtain the series form of the function  V,, as where Bessel functions are indispensable in many branches of mathematics and applied mathematics.Thus, it is important to study their properties in many aspects.We consider the generalized Dini functions  ,,, : U → C defined as where  ∈ R + ,  ∈ R,  ∈ C, and V ∈ R. For  =  =  = 1, we obtain the normalized Dini function of order V of the form By putting V = 1/2 and V = 3/2, we obtain the following particular cases of normalized Dini functions Recently Baricz et al. [13] studied the close-to-convexity of Dini functions and some monotonicity properties and functional inequalities for the modified Dini function are discussed in [5].Further some geometric properties of Dini functions are studied in [14].
This paper studies the Dini function  V,,, given by the power series (14).We determine the conditions on parameters that ensure the Dini function to be star-like of order , convex of order , and close-to-convex of order ((1 + )/2).We also study the convexity in the domain U 1/2 = { : || < 1/2}.Sufficient conditions on univalency of an integral operator defined by Dini function are also studied.We find the conditions on normalized Dini function to belong to the Hardy space H  .
To prove our main results, we need the following lemmas.
Lemma 2 (see [16]).Let  ∈ C with Re() > 0, and then the integral operator is analytic and univalent in U.
Putting  = 0 in Theorem 6, we have the following results.
Corollary 8. Let  ∈ R + ,  ∈ R,  ∈ C, and V ∈ R. Then the following assertions are true.
Consider the integral operator F  : U → C, where  ∈ C,  ̸ = 0, Here F  ∈ A. In the next theorem, we obtain the conditions so that F  is univalent in U.
then F  is univalent in U.

Proof. A calculation gives us
Since  ,,, ∈ A, then, by the Schwarz Lemma, triangle inequality, and (29), we obtain (50) This shows that the given integral operator satisfies Becker's criterion for univalence, and hence F  is univalent in U.

Strong Starlikeness
In this section, we are mainly interested about some sufficient conditions under which the normalized Dini function belongs to the class of strong starlikeness of order .

Close-to-Convexity with respect to Certain Functions
Recently many authors discuss the close-to-convexity of some special functions with respect to certain functions.Here, we are also interested in the close-to-convexity of normalized Dini function.
To prove that  is close-to-convex with respect to the function −log(1−), we use Lemma 5. Therefore, we have to prove that { −1 } ≥2 is a decreasing sequence.After some computations, we obtain (61) By using the conditions on parameters, we easily observe that  −1 − ( + 1)  > 0 for all  ≥ 2, and thus { −1 } ≥2 is a decreasing sequence.By Lemma 5 it follows that  is close-to-convex with respect to the function − log(1− ).

Let
H denote the class of all analytic functions in the open unit disk U = { : || < 1} and H ∞ denote the space of all bounded functions on H.This is a Banach algebra with respect to the norm By using Theorem 23 and Corollary 25, then we have the following corollary.Corollary 27.If  ∈ R, then the convolutions  1/2 *  and  3/2 *  are in H ∞ ∩ R.Moreover, if  ∈ R(1/2), then  1/2 *  and  3/2 *  ∈ R(0).