Bounds for the second Hankel determinant of certain univalent functions

The estimates for the second Hankel determinant a2a4−a32 of the analytic function f(z)=z+a2z2+a3z3+⋯ , for which either zf′(z)/f(z) or 1+zf″(z)/f′(z) is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained. MSC:30C45, 30C80.

Hankel determinants are useful, for example, in showing that a function of bounded characteristic in D, i.e., a function which is a ratio of two bounded analytic functions, with its Laurent series around the origin having integral coefficients, is rational [7].For the use of Hankel determinant in the study of meromorphic functions, see [40], and various properties of these determinants can be found in [38,Chapter 4].In 1966, Pommerenke [32] investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as for starlike functions.In [33], he proved that the Hankel determinants of univalent functions satisfy (n = 1, 2, . .., q = 2, 3, . ..), where β > 1/4000 and K depends only on q.Later, Hayman [15] proved that |H 2 (n)| < An 1/2 , (n = 1, 2, . . .; A an absolute constant) for areally mean univalent functions.In [21][22][23], the estimates for Hankel determinant for areally mean p-valent functions were investigated.ElHosh obtained bounds for Hankel determinants of univalent functions with positive Hayman index α [9] and of k-fold symmetric and close-to-convex functions [10].For bounds on the Hankel determinants of close-to-convex functions, see [24][25][26].Noor studied the Hankel determinant of Bazilevic functions in [27] and of functions with bounded boundary rotation in [28][29][30][31].In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [5,[12][13][14]16,[18][19][20].The Hankel determinant H 2 (1) = a 3 − a 2 2 is the well known Fekete-Szegö functional.For results The bounds for the second Hankel determinant H 2 (2) = a 2 a 4 − a 2 3 are obtained for functions belonging to these subclasses of Ma-Minda starlike and convex functions in Section 2. In section 3, the problem is investigated for two other related classes defined by subordination.In proving our results, we do not assume the univalence or starlikeness of ϕ as they were required only in obtaining the distortion, growth estimates and the convolution theorems.The classes introduced by subordination naturally include several well known classes of univalent functions and the results for some of these special classes are indicated as corollaries.
Let P be the class of functions with positive real part consisting of all analytic functions p : D → C satisfying p(0) = 1 and Re p(z) > 0. We need the following results about the functions belonging to the class P: LEMMA 1.1.[8] If the function p ∈ P is given by the series then the following sharp estimate holds: LEMMA 1.2.[11] If the function p ∈ P is given by the series (1.2), then for some x, z with |x| ≤ 1 and |z| ≤ 1.

Second Hankel determinant of Ma-Minda starlike/convex functions
Various subclasses of starlike functions are characterized by the quantity z f ′ (z)/ f (z) lying in some domain in the right half-plane.For example, f is strongly starlike of order β if z f ′ (z)/ f (z) lies in a sector | arg w| < β π/2 while it is starlike of order α if z f ′ (z)/ f (z) lies in the half-plane Re w > α.The various subclasses of starlike functions were unified by subordination in [17].The following definition of the class of Ma-Minda starlike functions is the same as the one in [17] except for the omission of starlikeness assumption of ϕ.
The class S * (ϕ) of Ma-Minda starlike functions with respect to ϕ consists of functions f ∈ A satisfying the subordination For the function ϕ given by ϕ α (z Then the class is the parabolic starlike functions introduced by Rønning [34].For a survey of parabolic starlike functions and the related class of uniformly convex functions, see [3].For 0 < β ≤ 1, the class is the familiar class of strongly starlike functions of order β .The class is the class of lemniscate starlike functions studied in [37]. THEOREM 2.1.Let the function f ∈ S * (ϕ) be given by (1.1).
(1) If B 1 , B 2 and B 3 satisfy the conditions (2) If B 1 , B 2 and B 3 satisfy the conditions ).
(3) If B 1 , B 2 and B 3 satisfy the conditions PROOF.Since f ∈ S * (ϕ), there exists an analytic function w with w(0) = 0 and |w(z Define the functions p 1 by Then p 1 is analytic in D with p 1 (0) = 1 and has positive real part in D. By using (2.3) together with (2.1), it is evident that 2), (2.4) and (2.5) that Therefore

Let
(2.6) , Since the function p(e iθ z) (θ ∈ R) is in the class P for any p ∈ P, there is no loss of generality in assuming c 1 > 0. Write . Substituting the values of c 2 and c 3 respectively from (1.4) and (1.5) in (2.7), it follows that Replacing |x| by µ and substituting the values of d 1 , d 2 , d 3 and d 4 from (2.6), yield ≡ F(c, µ).

Also note that
where P, Q, R are given by (2.10).( 2. Let ϕ : D → C be analytic and ϕ(z) is given as in (2.1).The class C (ϕ) of Ma-Minda convex functions with respect to ϕ consists of functions f satisfying the subordination THEOREM 2.2.Let the function f ∈ C (ϕ) be given by (1.1).

DEFINITION 2 . 1 .
Let ϕ : D → C be analytic and the Maclaurin series of ϕ is given by(2.1)