Convexity Properties for Certain Classes of Analytic Functions Associated with an Integral Operator BenWongsaijai and

and Applied Analysis 3 where (f ∗ g)(z)/z ̸ = 0, f(z) = z + ∑∞ j=1 a p+j z , and


Introduction
In the field of geometric function theory, the class of univalent functions [1,2] has been mainly studied.There are many distinguished geometric properties that played important role in the theory of univalent functions, such as starlikeness, convexity, and close-to-convexity (see, e.g., [3][4][5]).One of the generalizations of univalent functions is the theory of multivalent functions or -valent functions.Also, the geometric properties for the subclasses of -valent functions 2 Abstract and Applied Analysis By using the Alexander-type criterion, it follows that  () ∈ K  () ⇐⇒   ()  ∈ S *  () .
A function  ∈ US  (, ) is said to be -uniformly valent starlike of order  (−1 ≤  < ,  ≥ 0) in Both US  (, ) and UK  (, ) are comprehensive classes of analytic functions that include various classes of analytic univalent functions as well as some very well-known ones.For example, in the case  = 1, we have US 1 (, ) ≡ US(, ) and UK 1 (, ) ≡ UK(, ) which are introduced by Bharati et al. [12].For  = 0, UK 1 (, 0) ≡ UK() is the class of -uniformly convex function [13].In the special case,  =  = 1 and  = 0, the class US 1 (1, 0) ≡ US of uniformly starlike functions and UK 1 (1, 0) ≡ UK of uniformly convex functions were introduced by Goodman [14,15].Using the Alexander type relation, statement (7) holds for the UK  (, ) and US  (, ); that is, Many researchers have studied the geometric properties of integral operators.The common investigation is finding sufficient conditions of integral operators in order to transform analytic functions into classes with each of those mentioned properties.The well-known integral transformation defining a subclass of univalent functions was introduced by Alexander in [16].It is of the following form: In [17], Kim and Merkes extended the integral operator (12) by introducing a complex parameter  as Another object of investigation for the studies of the integral operator by Pfaltzgraff [18] is   defined by Until now, the various generalized form of the integral operators   in (13) and   in ( 14) has been investigated.However, Breaz and Stanciu [19] introduced and studied the more general form of integral operator  ,  : By setting appropriate values for the parameters , , and , integral operators that have been previously introduced can be obtained.In particular, if   = 0, then the integral operator  ,0  becomes the integral operator    introduced by D. Breaz and N. Breaz [20].Also, when   = 0, the integral operator  0,  is exactly the integral operator    defined by Breaz et al. [21].The specialized form of    and    involving the Bessel functions was introduced and studied in [22][23][24].In addition, the specific case  = 1 for  ,  in (15),  , 1 =  , was investigated by Pescar in [25].The univalence and their properties of the integral operators are reported in [26][27][28][29][30].
In [31], Bulut developed the integral operator  ,  : A   × A   → A  which extends the class of analytic functions A to the class of -valent functions A  ; that is, By setting   = 0 and   = 0, we obtain the integral operators  ,0 , =   , and  0, , =   , , respectively, which were introduced by Frasin [32].Also, some properties of these integral operators have been studied in [32][33][34].
In fact, we can write the Hadamard product in the form ( * )() =   ℎ() where ℎ is analytic in D and ℎ(0) = 1.Indeed, which usually appears in most of integral operators and always belongs to the class A 0 .That is why we are interested in replacing the term (  *   )()/  in the integral operator (19) with general function in A 0 .Additionally, replacing the term  −1 with a function () ∈ A −1 yields the integral operator, which is still contained in A  .We now define the following general integral operator where ℎ  ∈ A 0 for all  = 1, 2, . . ., .
The main purpose of the paper is to investigate the sufficient conditions on convexity of the integral operator I   [ℎ] on classes S *  (), S * (, ), and US(, ) of analytic functions.Our main results will be applied to reinstate the results of former researches with related integral operators.

Main Results
In this section, we investigate sufficient conditions for the convexity of the integral operator I   [ℎ] which is defined by (22).For the convenience, we introduce the transformation operator T  : where  ∈ A  and  is a nonnegative integer.In particular, we set T 1 = T.
We now prove a general property which guarantees the convexity of the proposed integral operator on the class S *  ().
Using the same method and technique as that in Theorem 1 with the nonnegativity of modulus of complex numbers, we are led easily to Theorem 2. The proof is omitted.Theorem 2. Let T() ∈ US  (, ) and T  (ℎ  ) ∈ US  (  ,   ) for  = 1, 2, . . ., .
The following is a result on the transformation property of I   [ℎ] on the class S * (, ).
then the integral operator I   [ℎ] defined by (22) is in the class K(,  − ∑  =1   ( −   )).Furthermore, Proof.From (26), we obtain That is, Theorem 3 can be applied to the integral operator Over the past few decades, there are many studies on the sufficient conditions that make the integral operators univalent.In fact, the class of convex functions is a subclass of the class of all univalent functions in D. Thus, it is interesting to observe that many results on the univalence property of integral operators follow the convexity property according to main results, especially Theorem 3 or Remark 4.
We now consider the integral operator   : A  × A  → A defined in (17).In order to obtain the convexity of the integral operator   by Theorem 3 or Remark 4, we set () = 1,  = 1,   = 0,  = ( 1 ,  2 , . . .,   ) ∈ C  , and where   ,   ∈ A. Other than that, the univalent property of   is also obtained.This implies Theorem 3.1 of Frasin in [39].Moreover, Frasin [39] noticed that for suitable functions   ∈ A, the integral operator   generalizes many operators introduced by several authors, for instance, Theorem 1 in [20], Theorem 2.1 in [41], Theorem 2.3 in [42], and Theorem 2.3 in [43].It is noteworthy to say that, under same assumptions, the former researches obtain only the univalence, while we obtain the stronger result, which is the convexity.
Our results can be used to explain the convexity of the other integral operators that are related to the Hadamad product as described next.
where   ,   ∈ A  .Then, we can apply the main results to discuss the convexity of the integral operator  , defined by (19).
The above statement also holds for the pairs of classes US  (, ) − UK  (, ) and S(, ) − K(, ).