On the Zalcman conjecture for starlike and typically real functions

Generalized Zalcman Conjecture for Starlike and Typically Real Functions

“…for f(z) = z + a 2 z 2 + a 3 z 3 + E S. This conjecture implies the famous Bieberbach conjecture lanl I n (see [2]) and is known to be true for n = 2 [3]. Results on various subclasses of S do support the conjecture.…”

“…We denote by S* the subclass of normalized ( f ( 0 ) =f'(O) - The class of all normalized typically real functions is denoted by T. The Zalcman conjecture is known to be true for the subclass S, consisting of functions in S with real coefficients (see [2]). Brown and Tsao [2] proved the Zalcman conjecture for S*.…”

The Zalcman Conjecture for Close-to-Convex Functions

“…In In this paper, we prove the Zalcman conjecture for the class C of close-to-convex functions and n > 4. Thus the result in [1] on the starlike functions is an immediate consequence of ours when n > 4. This provides further evidence in favour of the Zalcman conjecture.…”

“…In [1], the authors pointed out that the Zalcman conjecture is false in HS* for n = 2 and 3, so it is also false in HC for n = 2 and 3. For n = 2, the conjecture is true in 5, so also in C. However, the conjecture remains open in C for n = 3.…”

The Zalcman conjecture for close-to-convex functions

“…Recently, Zalcman conjectured that (1) |a2-a2n_i|<(n-l)2 for each n > 2 and each f(z) = z + a2z2 + • ■ ■ G S, which implies the famous Bieberbach conjecture |an| < n [1]. It is well known that (1) is valid for n = 2 [2].…”