On a Certain Subclass of Analytic Functions Involving Integral Operator Deﬁned by Polylogarithm Function

: In the present paper, we have introduced a new subclass of analytic functions involving integral operator deﬁned by polylogarithm function. Necessary and sufﬁcient conditions are obtained for this class. Further distortion theorem, linear combination and results on partial sums are investigated.


Introduction
The class A consists of functions of the form f (z) = z + ∞ ∑ k=2 a k z k (1) which are analytic in the unit disc U = {z : |z| < 1}. Let S denote the subclass of A, which consists of functions of the form (1) that are univalent and normalized by the conditions f (0) = 0 and f (0) = 1 in U. Many authors have investigated the properties of subclasses of S and their results have several applications in engineering, hydrodynamics and signal theory. Some of their results have to do with the starlikeness and convexity properties of subclasses of univalent functions S. One of the significant problems in geometric function theory are the extremal problems, which create an effective method for finding the existence of analytic functions with certain natural properties.
Extremal problems play an important role in geometric function theory, for finding coefficient bounds, sharp estimates, and an extremal function. The theory of analytic univalent functions is a powerful tool in the study of many problems related to the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection. The results we obtained here may have prospective application in other branches of mathematics, both pure and applied.
In addition, in [1] Silverman introduced the class T of analytic functions with negative coefficients consisting of functions f of the form In this section we obtain a sufficient condition for a function f given by (1) to be in Φ δ c (λ, β) and we prove that it is also a necessary condition for a function belonging to the class TΦ δ c (λ, β). Also, distortion results and linear combinations for the class TΦ δ c (λ, β) are obtained. We also investigate the results on partial sums for the functions in the class Φ δ c (λ, β).

Conditions for Functions to Be in the Class
Proof. Since 0 ≤ β < 1 and λ ≥ 0, now if we put . This implies that the desired in equality (6) holds. If f (z) = z (|z| = r < 1), then there exist a coefficient 1+c k+c δ a k = 0 for some k ≥ 2. It follows that Further note that By coefficeint inequality (6), we obtain Hence we obtain Then f ∈ Φ δ c (λ, β). This completes the proof.
In the next theorem, we prove that the condition (6) is also necessary for a function f ∈ TΦ δ c (λ, β).

This implies that
Noting that Now and consequently by (8) we get Letting r → 1, we get This proves the converse part.

Remark 1.
If a function f of the form (2) belongs to the class TΦ δ c (λ, β) then The equality holds for the functions Next we obtain the distortion bounds for functions belonging to the class TΦ δ c (λ, β).
If the sequence is nondecreasing, then The result is sharp. The extremal function is the function f 2 of the form (9).
Proof. Since f ∈ TΦ δ c (λ, β), we apply Theorem 2 to obtain Also we have, k+c δ r 2 , and (10) follows. In similar manner for f , the inequalities k+c δ are satisfied, which leads to (11). This completes the proof.
Then f ∈ TΦ δ c (λ, β) if and only if f can be expressed in the form and ∞ ∑ k=1 µ k = 1.

Proof. If a function f is of the form
which provides (7), hence f ∈ TΦ δ c (λ, β) by Theorem 2.
Conversely, if f is in the class f ∈ TΦ δ c (λ, β), then we may set and Then the function f is of the form (13) and this completes the proof.

Partial Sums of Functions in the Class
For a function f ∈ A given by (1), Silverman [12] investigated the partial sums f 1 and f m defined by In [12], Silverman examined sharp lower bounds on the real part of the quotients between the normalized convex or starlike functions and their sequences of partial sums. Also, Srivastava et al. [13] and Owa et al. [14] have investigated an interesting results on the partial sums. In this section, we consider partial sums of functions in the class Φ δ c (λ, β) and obtain sharp lower bounds for the ratios of real part of f to f m and f to f m .

Theorem 4.
Let a function f of the form (1) belong to the class Φ δ c (λ, β) and satisfy (6). Then where Thus by Theorem 1 we get, it suffices to show that Re(g(z)) > 0, z ∈ U. Applying (18) we find that which gives, and the proof is complete.

Conclusions
The theory of analytic function is an old subject, yet it remains an active field of current research. As a preferential topic concerning inequalities in complex analysis, there have been lots of studies based on the classes of analytic functions. The interplay of geometry and analysis is the most fascinating aspect of complex function theory. This rapid progress has been concerned primarily with such relations between analytic structure and geometric behavior. Motivated by this approach, in the present study, we have introduced a new subclass of analytic functions involving integral operator defined by polylogarithm function. Necessary and sufficient conditions are obtained for this class. Further distortion theorem, linear combination and results on partial sums are investigated in this study, and therefore it may be considered as a useful tool for those who are interested in the above-mentioned topics for further research.