INTEGRAL MEANS AND DIRICHLET INTEGRAL FOR ANALYTIC FUNCTIONS MILUTIN OBRADOVI ´ C, SAMINATHAN PONNUSAMY † , AND KARL-JOACHIM WIRTHS Abstract. For normalized analytic functions f in the unit disk, the estimate of the integral means L 1 (r, f ) := r 2 2π ∫ π -π dθ |f (re iθ )| 2 is important in certain problems in fluid dynamics, especially when the functions f (z) are non-vanishing in the punctured unit disk 0 < |z| < 1. We consider the problem of finding the extremal function f which maximizes the integral means L 1 (r, f ). In addition, for certain class F of analytic functions, we solve the ex- tremal problem for the Yamashita functional A(r) = max f ∈F ∆ ( r, z f (z) ) for 0 <r ≤ 1. 1. Preliminaries and Main Results Denote by H the family of all functions f which are analytic in the unit disk D := {z ∈ C : |z | < 1} and by A the subfamily of H with the normalization f (0) = 0= f ′ (0) − 1. Also, let S = {f ∈A : f is univalent in D} and S ∗ := S ∗ (0) ⊂S denote the class of all starlike (univalent) functions in D. Here S ∗ (β ) denotes the family of starlike functions of order β , i.e., functions f ∈S such that [7] Re ( zf ′ (z ) f (z ) ) > β, z ∈ D, where 0 ≤ β< 1. For f ∈H, the integral means I 1 (r, f ) := 1 2π ∫ π −π dθ |f (re iθ )| 2 and the estimates of I 1 are important in certain problems in fluid dynamics (see [8, 22, 23]). Recently the authors in [16] obtained that if f ∈S ⋆ (β ), then the estimate L 1 (r, f ) := r 2 I 1 (r, f ) ≤ Γ(5 − 4β ) Γ 2 (3 − 2β ) 2000 Mathematics Subject Classification. Primary: 30C45, 30C70; Secondary: 30H10, 33C05. Key words and phrases. Analytic, univalent, Hadamard product, starlike functions, Dirichlet- finite, area integral. † This author is on leave from the Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India . File: Samy-W5(13)˙final.tex, printed: 18-10-2013, 16.10. 1