Coefficient inequality for transforms of parabolic starlike and uniformly convex functions

The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the k root transform [ f(z) ] 1 k of normalized analytic function f(z) belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.


Introduction
Let A denote the class of all functions f (z) of the form f (z) = z + ∞ n=2 a n z n (1.1) in the open unit disc E = {z : |z| < 1}. Let S be the subclass of A consisting of univalent functions. Let the functions F and G be analytic in the unit disc E. Then F is said to be subordinate to G, written F ≺ G, if there exists an analytic function w(z) in the open unit disc E satisfying w(0) = 1 and |w(z)| < 1, ∀z ∈ E called the Schwarz's function such that If F ≺ G and G(z) is univalent in the open unit disc E, then the subordination is equivalent to F (0) = G(0) and range F (z) ⊆ range G(z). For a univalent function in the class A, it is well known that the n th coefficient is bounded by n. The bounds for the coefficients give information about the geometric properties of these functions. For example, the bound for the second coefficient of normalized univalent function readily yields the growth and distortion properties for univalent functions. The Hankel determinant of f for q ≥ 1 and n ≥ 1 was defined by Pommerenke [18] as H q (n) = a n a n+1 · · · a n+q−1 a n+1 a n+2 · · · a n+q . . . . . . . . . . . .
This determinant has been considered by many authors in the literature . For example, Noor [17] determined the rate of growth of H q (n) as n → ∞ for the functions in S with bounded boundary. Ehrenborg [8] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence and some of its properties were discussed by Layman in [13]. In 1966, Pommerenke [18] investigated the Hankel determinant of areally mean p−valent functions, also studied by Noonan and Thomas [16], univalent functions as well as starlike functions. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [1,12,11]. In particular cases, q = 2, n = 1, a 1 = 1 and q = 2, n = 2, a 1 = 1, the Hankel determinant simplifies respectively to We refer to H 2 (2) as the second Hankel determinant. It is fairly well known that for the univalent functions of the form given in (1.1) the sharp inequality H 2 (1) =| a 3 − a 2 2 |≤ 1 holds true [7]. For a family T of functions in S, the more general problem of finding sharp estimates for the functional | a 3 − µa 2 2 | (µ ∈ R or µ ∈ C) is popularly known as the Fekete-Szegö problem for T . Ali [3] found sharp bounds for the first four coefficients and sharp estimate for the Fekete-Szegö functional | γ 3 −tγ 2 2 |, where t is real for the inverse function of f defined as f −1 (w) = w + ∞ n=2 γ n w n when it belongs to the class of strongly starlike functions of order α (0 < α ≤ 1) denoted by ST (α). R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam [5] obtained sharp bounds for the Fekete-Szegö functional denoted by | b 2k+1 − µb 2 k+1 | associated with the k th root transform f (z k ) 1 k of the function given in (1.1), belonging to certain subclasses of S. The k th root transform for the function f given in (1.1) is defined as Motivated by the results obtained by R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam [5], in the present paper, we obtain sharp upper bound to the functional | t k+1 t 3k+1 − t 2 2k+1 |, called the second Hankel determinant for the k th root transform of the function f when it belongs to certain subclasses of S, defined as follows.
The class of all parabolic starlike functions is introduced by Ronning [20] and is denoted by S p . Geometrically, (see [4]) S p is the class functions f , for which zf (z) f (z) takes its value in the interior of the parabola in the right half plane symmetric about the real axis with vertex at ( 1 2 , 0). Definition 1.2. A function f ∈ A is said to be in U CV , if and only if (1.5) Goodman [9] introduced the class U CV of uniformly convex functions consisting of convex functions f ∈ A with the property that for every circular arc γ contained in the unit disc E with centre also in E, the image arc f (γ) is a convex arc. Ma and Minda [15] and Ronning [20] independently developed a one-variable characterization for the functions in the class U CV . From the Definitions 1.1 and 1.2, we have the relation between U CV and S p is given in terms of an Alexander type Theorem [2] by Ronning (see [4]) as follows.
(1.6) Further, Ali [4] obtained sharp bounds on the first four coefficients and Fekete-Szegö inequality for the functions in the class S p . Ali and Singh [6] showed that the normalized Riemann mapping function q(z) from E onto the domain denotes the parabolic region in the right half plane of the complex plane given by Some preliminary lemmas required for proving our results are as follows:

Preliminary Results
Let P denote the class of functions consisting of p, such that and c −k = c k , are all non-negative. They are strictly positive except for This necessary and sufficient condition found in [10] is due to Caratheòdory and Toeplitz. We may assume without restriction that c 1 > 0. On using Lemma 2.2, for n = 2, we have which is equivalent to For n = 3, and is equivalent to 3) From the relations (2.2) and (2.3), after simplifying, we get To obtain our results, we refer to the classical method initiated by Libera and Zlotkiewicz [14] and used by several authors in the literature.

Main Results
and the inequality is sharp.
Proof. For f (z) = z + ∞ n=2 a n z n ∈ S p , by virtue of Definition 1.1, we have Using the series representations for f (z), f (z) and h(z) in (3.3), we have Upon simplification, we obtain Equating the coefficients of like powers of z, z 2 and z 3 respectively on both sides of (3.5), after simplifying, we get where p(z) is given in (2.1). Solving w(z) in terms of p(z) in the relation (3.7) and replacing p(z) by its equivalent expression in series, we have Upon simplification, we obtain .. for q(z) given in (1.7), after simplifying, we get From the relations (1.7) and (3.9), we obtain Equating the coefficients of like powers of z, z 2 and z 3 respectively, on both sides of (3.10), we get In view of (3.12), using (3.8) in (3.10) along with the equivalent expression for h(z) given in (3.3), upon simplification, (3.12) is equivalent to Equating the coefficients of like powers of z, z 2 and z 3 respectively, on both sides of (3.13), we have Simplifying the relations (3.11) and (3.14), we get From the relations (3.6) and (3.15), upon simplification, we obtain For a function f given by (1.1), a computation shows that Substituting the values of t k+1 , t 2k+1 and t 3k+1 from (3.18) in the second Hankel determinant | t k+1 t 3k+1 − t 2 2k+1 | to the k th transformation for the function f ∈ S p , upon simplification, we obtain Using the triangle inequality and the fact that |z| < 1, we get From the relation (3.21), we can now write (d 1 + 2d 2 + d 3 + 4d 4 ) = 32 15k 2 π 4 k 2 π 4 − 360 ; d 1 = 12; ( Since c 1 ∈ [0, 2], using the result (c 1 + a)( Substituting the calculated values from (3.24) and (3.26) on the right-hand side of (3.23), we have Choosing c 1 = c ∈ [0, 2], applying triangle inequality and replacing |x| by µ on the right-hand side of (3.27), we get Using these values in (3.35), we observe that equality is attained, which shows that our result it sharp. For these values, we derive that Therefore, in this case the extremal function is zf (z) This completes the proof of our Theorem 3.1.

Theorem 3.2. If f given by (1.1) belongs to U CV and F is the k th root transformation of f given by (1.3) then
and the inequality is sharp.