The Zalcman conjecture for close-to-convex functions

Abstract

Let S S be the class of functions f ( z ) = z + ⋯ f(z) = z + \cdots analytic and univalent in the unit disk D D . For f ( z ) = z + a 2 z 2 + ⋯ ∈ S f(z) = z + {a_2}{z^2} + \cdots \in S , Zalcman conjectured that | a n 2 − a 2 n − 1 | ≤ ( n − 1 ) 2 ( n = 2 , 3 , … ) |a_n^2 - {a_{2n - 1}}|\; \leq \;{(n - 1)^2}(n = 2,3, \ldots ) . This conjecture is verified for n ≥ 4 n \geq 4 and close-to-convex functions.

On the generalized Zalcman functional $\lambda a_n^2-a_{2n-1}$ in the close-to-convex family

“…In 1988, Ma [16] provided further evidence in support of the conjecture by verifying the Zalcman inequality (2) (again for the case λ = 1 only) for the class C when n ≥ 4. However, the conjecture remains open in C for n = 3 (see [16,Remarks]).…”

“…In fact they have proved it in a general form involving generalized Zalcman functional, and another generalization of the result of Brown and Tsao appeared in [17]. In 1988, Ma [16] provided further evidence in support of the conjecture by verifying the Zalcman inequality (2) (again for the case λ = 1 only) for the class C when n ≥ 4. However, the conjecture remains open in C for n = 3 (see [16,Remarks]).…”

Generalized Zalcman conjecture for convex functions of order α

“…of f ∈ S with the power series of (1), we find that Sharp bound for the generalized Fekete-Szegő functional has been established for several subclasses of S (see [4,16,17]) and more recently in [1,14,15]. The Zalcman coefficient inequality for n = 3 and for the full class S, was established in [12] and also for the special cases n = 4, 5, 6 in [13].…”

On coefficient functionals associated with the Zalcman conjecture

“…The problem (1.1) has been studied for several well-known subclasses of the class S. For example, in [5], Brown and Tsao proved that (1.1) holds for the class T of typically real functions and the class S * of starlike functions. In [17], Ma proved the Zalcman conjecture for the class K of close-to-convex functions when n ≥ 4. Readers can refer to, for instance, [1,12,13,14] and references therein for more information on this topic.…”