THIRD ORDER HANKEL DETERMINANT FOR CERTAIN UNIVALENT FUNCTIONS

By Deepak Bansal, S. Maharana, Jugal Kishore Prajapat. 2015

99 citations recorded by the source corpus.

Abstract. The estimate of third Hankel determinantH 3,1 (f) =a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 of the analytic function f(z) = z+ a 2 z 2 + a 3 z 3 + ··· , for which ℜ(1 +zf ′′ (z)/f ′ (z)) >−1/2 are investigated. The corrected version of a knownresults [2, Theorem 3.1 and Theorem 3.3] are also obtained. 1. IntroductionLet H(D) denote the class of analytic functions in the open unit disk D={z ∈ C: |z| 0, z ∈ D. Functions in R are known to be close-to-convex (andhence univalent) in D. Further, a function f ∈ A is called starlike (with respectto the origin 0), if tw ∈ f(D) whenever w ∈ f(D) and t ∈ [0,1]. We denote byS ∗ the subclass of A whose members are starlike in D. It is well known thatf ∈ S

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b2ec552090634d53a66a07416c13d8f3e93db8aa

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In Applied Mathematics

1. Dorina Raducanu
2. Pawel Zaprawa
3. Hari M. Srivastava

In Geometry and Topology

1. Dorina Raducanu
2. Pawel Zaprawa
3. Nanjundan Magesh
4. Hari M. Srivastava
5. J. Yamini
6. Halit Orhan

Citations in the corpus

• THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS (2018, unknown): “…triangle inequality, the formula (1.3) yields This simple observation allows one to estimate | H 3 , 1 ( f ) | for compact subclasses F of A (see [1, 2, 18–20 , 22 having a positive real part in D , the coe ffi cients of functions in F can be usefully represented by coe ffi cients of functions in P .”
• Second Hankel determinant for close-to-convex functions (2017, background): “…for example, [1], [7], [8], [13], [22], [23]) There are, however, few papers that discuss the third Hankel determinant H ( 1 ) (see, for example: [2], [21], [24]) Although many estimates of | H ( 2 ) | are sharp, for example for the classes S ∗ or K consisting of starlike or convex functions,…”
• SHARP BOUNDS OF SOME COEFFICIENT FUNCTIONALS OVER THE CLASS OF FUNCTIONS CONVEX IN THE DIRECTION OF THE IMAGINARY AXIS (2019, unknown): “Many authors have examined the second Hankel determinant H 2 , 2 ( f ) = a 2 a 4 − a 23 (see, for example, [2, 4, 5, 10, 11, 17, 21]).”