Fractional Calculus for Certain Integral Operator Involving Logarithmic Coefficients 1

Problem statement: Some properties of certain integral operators on some subclasses were studied. Approach: Certain classes defined by integral operators were introduced. The well known definitions and preliminaries results were stated. Results: Having new integral operator, the characterization problems were discussed. Thus sufficient conditions were given. Conclusion: Therefore, by having new integral operators, sufficient conditions were determined. In fact, other properties from this class could be obtained.


INTRODUCTION
The study of integral operators has been rapidly investigated by many authors in the field of univalent functions. Recently, various integral operators have been introduced for certain class of analytic univalent functions in the unit disk. In this study, we follow the similar approach by introducing a logarithmic coefficients of analytic functions in the punctured disk. We begin by giving some well-known notations and preliminary results on the class defined by integral operators and also the basic knowledge of logarithmic. Later we derive the integral operator aforementioned. Once the integral operator being derived, we shall discuss on the sufficient conditions of certain classes defined.
Let denote by A the class of functions f normalized by: We also denote by S the subclass of A consisting of functions which are also univalent in U. Furthermore, we denote by T the subclass of S consisting of functions whose nonzero coefficients, from the second one, are negative and normalized by: Associated with each function f in S are its logarithmic coefficients γ n defined by: The numbers γ n are called the logarithmic coefficients of f [9] . We next define the following fractional calculus (fractional integrals and fractional derivatives) given by Owa and Srivastava [7,8] .

Definition 2:
The fractional derivative of order λ is defined for f (z)∈A by: where, the multiplicity of (z ) −λ − ζ is removed by requiring (z ) − ζ to be real when (z ) 0 − ζ > . Therefore, we say that: for any real λ.

MATERIALS AND METHODS
Next we state two known definitions that lead to our definitions:

Definition 4:
A function f∈A is said to be in the class KD(µβ), if satisfies the following inequality [6] : for some µ≥0 and 0≤β< 1.

Definition 5:
A function f∈T is said to be in the class (K, A, B, µ) -UCV iff it satisfies the condition [5] : For some µ≥0 and 0≤β<1, where F m (z) is defined as in (1).

Theorem 1:
Let F m (z) be the integral operator defined by (1) then: and m n i n ,i i 1 n 1 where, z D λ and z D −λ are the fractional derivative and fractional integral respectively of f∈S and γ n (n = 1, 2,…) denote the logarithmic coefficients of f.
Since f i (z)∈S, then by using logarithmic coefficients γ n , we get: By using similar method we get the result (5).

DISCUSSION
Having classes defined previously, we first give a sufficient condition for a family of functions f i ∈KDF m (µ, β, α 1 ,…, α m ). Before embarking on the proof of our result, let us calculate the expression m m zF "(z) F '(z) , required for proving our result.
Then we have: m n 1 i n,i i 1 n 1 m n 1 i n,i i 1 m 1 If we choose z and ξ real and letting z→1and ξ→k + , we obtain:

CONCLUSION
The integral operator defined was motivated by Breaz and the group [1][2][3][4] This operator can be generalised further and many other results such as the coefficient estimates and distortion theorem can be obtained.