Let A denote the class of functions
which are analytic in the open unit disk and satisfy the condition and .
Let S denote the subclass of A consisting of univalent in U. A function is said to be starlike in the unit disk if and only if
Also, a function is said to be convex in the unit disk if and only if
Let be defined by
which is equivalent to
is the Salagean differential operator .
Fekete and Szegö  studied the estimate of a functional known as Fekete-Szegö functional, where is real. Also, Noonan and Thomas  defined the qth Hankel determinant of for by
This determinant has been considered for specific values q and n by many authors. It is well established that the Fekete-Szegö functional given by . Pommerenke  investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as starlike functions. Noor  investigated the Hankel determinant problem for the class of functions with bounded boundary rotation. Janteng et al.  studied the sharp upper bound for second Hankel determinant for univalent functions whose derivative has positive real parts. Also, Lee et al.  obtained bounds on second Hankel determinants belonging to the subclasses of Ma-Minda starlike and convex functions. Bansal  has obtained bounds on for a new class of analytic functions.
In this paper, we obtained the coefficient bound, Fekete-Szegö functional and second Hankel determinant for the functions belonging to the subclass .
Definition 1.1. A function of the form (1.1) analytic and univalent in U is said to be in the and if it satisfies the inequality
(1) For the class gives
studied in .
(2) For gives
investigated by .
For , the class gives
studied in .
2. Preliminary Lemmas
We need the following lemmas to prove our results.
Let P denote the class of Caratheodory functions.
which are analytic and satisfy and
Lemma 2.1. Let . Then
Lemma 2.2. Let , then for any real
Lemma 2.3. Let then
for some value of , such that and .
3. Main Results
Theorem 3.1. Let and .
Let , then by [1.4]
Comparing coefficients of (3.1) and (3.3) gives
Solving for the bounds of (3.4), (3.5), (3.6) and using lemma 2.1 give
Theorem 3.2. Let , then for any real number
Using (3.4) and (3.5) give
then using lemma (2.2) in (3.10) gives
then by lemma 2.2 we obtain
then using lemma 2.2 gives
Theorem 3.3 Let and
Using (3.4), (3.5) and (3.6) give
Suppose , and recall that , and assuming without restriction that . Then, using triangle inequality
Now, putting then
Differentiating partially with respect to in the closed interval
for , therefore is an increasing function. Hence, it attains maximum point at. Thus,
Now, the critical points occur at
but the maximum point occurring at [3.19] becomes
A subclass of analytic functions which generalize some well known subclasses of analytic and univalent functions was defined. The initial coefficients upper bounds, upper estimates for the Fekete-Szegö functional and the second Hankel determinants for the class were obtained. The study unifies existing results and obtains new results in geometric function theory. Future researches can be done to obtain the geometric properties by using Chebyshev polynomials.
The authors wish to thank the referees for their valuable suggestions that lead to improvement of the quality of the work in this paper.
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