Bounds of Hankel Determinant for a Class of Univalent Functions

The authors study the coefficient condition for the class ℌ𝛼 defined as the family of analytic functions 𝑓,𝑓(0)=0 and 𝑓′(0)=1, which satisfy ℜ[(1−𝛼)𝑓′(𝑧)


Introduction
Let A be the class of functions of the following form: a n z n , 1.1 which are analytic in the unit disc E {z : |z| < 1}, and let S be the subclass of A consisting of functions which are univalent in E. A function f ∈ A is said to be close to convex in the open unit disc E if there exists a convex function g not necessarily normalized such that For fixed real numbers α, let H α denote the family of functions f in A which satisfy

International Journal of Mathematics and Mathematical Sciences
In 2005, V. Singh et al. 1 established that, for 0 < α < 1, functions in H α satisfy f z > 0 in E and so are close to convex in E.
In 2 , Noonan and Thomas defined the Hankel determinant H q n of the function f for q ≥ 1 and n ≥ 1 by The determinant has been investigated by several authors with the subject of inquiry ranging from rate of growth of H q n as n → ∞, to the determination of precise bounds on H q n for specific q and n for some special classes of functions.In a classical theorem, Fekete and Szeg ö 3 considered the Hankel determinant of f ∈ S for q 2 and n 1 The well-known result due to them states that if f ∈ S, then where a 1 1 and μ is a real number.In the present paper, we obtain a sharp bound for

Preliminary Results
We denote by P the family of all functions p z given by for some x, z such that |x| ≤ 1 and |z| ≤ 1.

3.1
where α 0 0.4276891324 . . . is the root of the equation 10α 3 − 5α 2 12α − 5 0 and Proof.Since f ∈ H α , it follows from 1.3 that there exists a function p ∈ P such that Equating coefficients in 3.3 yields

3.4
International Journal of Mathematics and Mathematical Sciences Thus, we can easily establish that

3.7
International Journal of Mathematics and Mathematical Sciences 5 Differentiating F ρ , we get the following:

3.8
Using elementary calculus, one can show that F ρ > 0 for ρ > 0. It implies that F is an increasing function, and, thus, the upper bound for F ρ corresponds to ρ 1, in which case

3.10
Setting G c 0, since 0 ≤ c ≤ 2, we have where K α is given by 3.2 .This completes the proof of the Theorem.
Setting α 0 in above theorem, we get the following result of Janteng et al.

3.14
The result is sharp.

2
International Journal of Mathematics and Mathematical Sciences 3 n 1, 2, 3 . . .and c −k = c k are all nonnegative.They are strictly positive except for j for k / j; in this case, D n > 0 for n < m − 1 and D n 0 for n ≥ m.Lemma 2.2 See 5, 6 .Let p ∈ P .Then