Partial Sums of Some Fractional Operators of Bounded Turning

In this paper, several conditions are put in order to compose the sequence of partial sums Bm, Lm and Em of the fractional operators of analytic univalent functions B , L and Es,t of bounded turning which are bounded turning too.


Introduction:
In a general series, it is not easy to create a summing of all the preceding terms in the series since the series begins with = 1 and moves until infinity. But a geometric series contains many polynomials characteristics, making it handy to work with. In this paper, the way we get the sum of a geometric series is by partial sums. In geometric functions theory, both analytic univalent functions and partial sum polynomials can be used to introduce a new certain linear operator of squares. In addition, the certain values depend on the coefficients in such partial sums. There are numerous interesting advancements in the partial sums of analytic univalent functions and classes of bounded turning in the open unit disk. In (1) it was proved that the partial sums of the Liberal integral operator of functions of bounded turning are bounded turning too. Moreover, some special functions are associated with calculus operators of analytic univalent functions as in (2) and (3). Recently, the partial sums of some special functions have been studied by authors for example (4,5) in the domain of open unit disk.
The main results are displayed using the method of partial sums of functions class of bounded turning, we review in this paper some definitions of functions type of bounded turning of analytic univalent and their properties. We begin with a definition of analytic functions and some geometric properties such as convolution (or Hadamard product), that is a binary operation of two or several analytic functions. Also, we consider three types of fractional (differential and integral) operators with their squares of partial sums in the open unit disk.

Preliminaries
Let . (2.1) Let the function be defined in (2.1), then the starlike functions are functions for which the real part of the quantity ( ′/ ) is positive and the convex functions are functions for which the real part of the quantity (1 + ′′ / ′ ) is positive. In contrast, the close-to-convex functions are functions for which the real part of the quantity ( ′/ℎ′ , ℎ is convex ) is positive. The following binary operation ( * ) symbolizes the convolution (or Hadamard product) of analytic functions in and is defined by where ( ) is defined in (2.1) and ℎ( ) = + ∑ , ∞ =2 for ∈ .

Definition 1 (6)
For 0 ≤ < 1, let in ( ) where ( ) is a subclass of , are known as the functions of bounded turning whose derivative contains positive real part and is denoted by close-to-convex function with ℎ( ) = in . For functions described in the equation (2.1), many families of analytic and univalent functions are considered (integral and differential) operators type of fractional, which have been introduced by numerous researchers for example (see (7),(8),(9),(10)). Moreover, the familiar fractional calculus operators of analytic and univalent functions are given in (2.1) which are respectively presented and studied by Srivastava and Owa (11) as follows: : where 1+ and ( ) are respectively in (2.2) and (2.3) the well known fractional differential operators that have been defined by Srivastava and Owa (11)

Results
By applying Lemma 1 and Lemma 2, we will demonstrate some conditions in order to compose -the partial sums of the operators (2.5), (2.6) and (2.7) of functions of bounded turning that are bounded turning as well.

Proof
The prove is similar in the sense to the proof Theorem 1.

Proof
From the hypotheses of the theorem, it is well known that Under the hypothesis ( + ) ≤ Γ( + ), , ∈ which is identical to , for (0.5 < < 1 ) and by this, the proof of Theorem 3 is completed.

Conclusions:
In this paper, we used the method of partial sums of functions class of bounded turning. The conditions of the partial sums of the fractional (differential and integral) operators are determined to be bounded turning too.