Estimates of the Second Derivative of Bounded analytic Functions

Abstract

Let z 0 and w 0 be given points in the open unit disk D with |w 0 | < |z 0 |, and H 0 be the class of all analytic self-maps f of D normalized by f (0) = 0. In this paper, we establish the third order Dieudonné Lemma, and apply it to explicitly determine the variability region {f ′′′ (z 0 ) : f ∈ H 0 , f (z 0 ) = w 0 , f ′ (z 0 ) = w 1 } for given z 0 , w 0 , w 1 and give the form of all the extremal functions.2010 Mathematics Subject Classification. Primary 30C80; secondary 30F45.

Variability regions for the second derivative of bounded analytic functions

“…Based on this result, the first author [3] gave the sharp estimate for |f ′′ (z 0 )|. In this paper, we would like to further study the second derivative f ′′ (z 0 ) by explicitly describing the variability region…”

Variability regions for the fourth derivative of bounded analytic functions

“…We appropriately modify Rivard's result as follows (see [2]). Denote c 2 (z 0 , w 0 , λ) and ρ 2 (z 0 , w 0 , λ) by…”