On the mean boundary behavior and the Taylor coefficients of an infinite Blaschke product (Q1077586)

The authors consider infinite Blaschke products B in the unit disc and establish the equality \[ \inf_{B} \limsup_{r\to 1}(1-r)^{- 1}\int^{\pi}_{-\pi}(1-| B(re^{\quad i\theta})|)^ 2 d\theta /2\pi = \] \[ \max_{0\leq x<1}(1+\sqrt{1-x})_ 2F_ 1(,2,;x)=\gamma_ 0, \] where \({}_ 2F_ 1(,2,;x)\) is a hypergeometric function. They apply this result to obtain information on the Taylor coefficients of infinite Blaschke products. Write \(B(z)=\sum^{\infty}_{n=0}a_ nz^ n\). Using an estimate of \(\gamma_ 0\), they establish that \[ 0.37749<\inf_{B}( \limsup_{n\to \infty}n | a_ n|)\leq 2/e, \] improving results of \textit{D. J. Newman} and \textit{H. S. Shapiro} [Mich. Math. J. 9, 249-255 (1962; Zbl 0104.296)]. They conjecture that equality holds on the right hand side.