On a Generalized Integral Operator

We have considered several integral operators from literature and we have made a generalization of them. It can be easily seen that their properties are also preserved. Therefore, we use known results concerning the starlike functions (see [1,2]) and we unify some known integral operators (see [3]) into one single integral operator, called I(z), in Section 3 of this paper.


Introduction
Let H(U) be the set of functions which are regular in the unit disc U, in 1999.Thus, we have the subfamily S-T consisting of functions f of the form   2 , 0, 2,3, , The purpose of this paper is to give a generalization with respect to several integral operators that exist in literature.In order to do so, we have firstly considered the univalent function f and the family S of analytic and univalent functions, notions that we use further.
for all z U, where the equality holds only if . Some integral operators and the related properties of them are also studied in [6,7].
The neighborhoods of the class of functions defined using the operator (3) is studied in [8].
Remark 2.2.In [9], we have introduced and studied the following operator of the functions f S, : [10], Prof. G. S. Sălăgean has introduced the subfamily T of S consisting of functions f of the Form (1); see Section 1.

Remark 2.3. If we denote by   k
x the Pochammer symbol, we define it as follows: .
Following, we introduce a new general integral operator in Theorem 3.1,   1 I z , giving also several examples which prove its relevance.We derive it as well in order to show its applicability.

Main Results
, , Then, for from where we obtain the following derivatives: 16) from hypothesis, we see that   1 P z  .By applying Schwarz Lemma, we have that By taking into account the inequalities (17), ( 19) and (20), we obtain the following: , , , , Re 0 a . We will use this form of the integral operator, where the function f is of Form (2) with respect to the operator (21).For further simplification, we consider that 1     and δ = 1 (except of Example 3.5).
For the first four examples we consider we obtain the operator F(z) of Form (7).

Conclusions
The integral operator which is introduced and studied in Theorem 3.1 of this paper, called I 1 (z), is a generalization of several integral operators that are taken from literature (for e.g., see [3]).Looking to the examples from above, we can easily see that our integral operator can be used further without any concern, because it preserves the initial state of the already known integral operators.Therefore, the integral operator I 1 (z) covers the integral operators from [3] without loss of generality.
Open Problem 3.1.It would be interesting to prove that the integral operator (18) or (21) covers also the integral operator from [5].
. S. Sălăgean has introduced the subfamily T of S consisting of functions f of the form

.
We denote by D   the linear operator defined by

means that the function   1 I 1 . 1 I
z of Form (18) is univalent for all The mapping properties with respect to integral operator   z of Form (18) are studied in[7].If we consider the operator   D z  of Form (3) we obtain the following Corollary.It can be proved in a similar way as the Theorem 3.1 is.Corollary 3.1.Let 1 2

Example 3 . 2 .Example 3 . 3 . 4 . 6 . 3 .
If σ = 1 we obtain the operator G(z) of Form (If σ = 1 and we use the notation .If σ = 0 we obtain the operator a) If χ = 1, δ = 1, we obtain a particular case of the function J(z) of Form (5), in which If n = 1 we obtain the operator H(z) of Form (It can be easily proved that the examples above are also true for f(z) of Form (1).