Absolute Cesàro summability and absolute convergence for starlike univalent functions (Q1078353)

We say that \(\sum a_ n e^{in\theta}\) is absolutely summable (C,\(\alpha)\), or summable \(| C,\alpha |\), if \(\sum | \sigma_ n^{(\alpha)}(\theta)-\sigma^{(\alpha)}_{n- 1}(\theta)|\) converges, where \(\sigma_ n^{(\alpha)}(\theta)\), \(\alpha\geq 0\), denotes the n-th Cesaro mean of the series. Summability \(| C,0|\) is equivalent to the convergence of \(\sum | a_ n|.\) In this note we prove, for example, that if \(f(z)=\sum^{\infty}_{1}a_ nz^ n\) is starlike univalent and bounded in \(U=\{z: | z| <1\},\) then, for each \(\theta\) in [0,2\(\pi\) ], \(\sum a_ n e^{in\theta}\) is \(| C,\alpha |\) summable for every \(\alpha >0\). We also show that there exists a bounded starlike univalent function \(f(z)=\sum a_ nz^ n\) having a continuous extension to \(\bar U,\) the closed unit disc, with boundary modulus of continuity function \(\omega\) satisfying \(\omega (\delta,f)=O((\log \delta^{-1})^{-1}),\) \(\delta\) \(\to 0\), for which \(\sum | a_ n| =\infty\). An earlier result of the author [J. Lond. Math. Soc., II. Ser. 29, 462-476 (1984; Zbl 0571.30002)] shows that \(\omega (\delta,f)=O((\log \delta^{-1})^{-\beta})\) for any \(\beta >1\) implies \(\sum | a_ n| <\infty\) for such functions.