Maximal area integral problem for certain class of univalent analytic functions

الأصل · EN

One of the classical problems concerns the class of analytic functions f on the open unit disk |z|<1 which have finite Dirichlet integral Δ(1,f), where Δ(r,f)=|z|<ᵣ|f'(z)|² dxdy (0<r≤ 1). The class S*(A,B) of normalized functions f analytic in |z|<1 and satisfies the subordination condition zf'(z)/f(z) (1+Az)/(1+Bz) in |z|<1 and for some -1≤ B≤ 0, A∈ C with A≠ B, has been studied extensively. In this paper, we solve the extremal problem of determining the value of f∈ S*(A,B)Δ(r,z/f) as a function of r. This settles the question raised by Ponnusamy and Wirths in [11]. One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradović et. al [9].