Fungsi $\alpha$-Cembung Secara Logaritma

Abstract

Untuk $\alpha>0,$ andaikan $M^\alpha$ sebagai kelas bagi fungsi ternormal cembung secara logaritma pada cakera unit $D$ dan diberi oleh \[\textrm{Ny}\left[\Big( 1 + \frac{zf''(z)}{f'(z)}\Big)^\alpha \Big(\frac{zf'(z)}{f(z)}\Big)^{1-\alpha}\right]>0,\] dan $f(z)^{1-\alpha}f'(z)^\alpha \ne 0.$ Untuk $f\in M^n$ dan diberi oleh \[f(z) = z + a_2z^2 + a_3 z^3 + ... ,\] batasan atas terbaik diperolehi bagi fungsian Fekete-Szeg\"{o} $\big|a_3-\mu a_2^2 \big|$ apabila $\mu$ nyata. Katakunci: Fungsi cembung; fungsi bak-bintang; fungsian Fekete-Szeg\"{o}; herotan. For $\alpha>0,$ let $M^\alpha$ be the class of normalised analytic -logarithmically convex functions defined in the open unit disc given by \[\textrm{Re}\left[\Big( 1 + \frac{zf''(z)}{f'(z)}\Big)^\alpha \Big(\frac{zf'(z)}{f(z)}\Big)^{1-\alpha}\right]>0,\] and $f(z)^{1-\alpha}f'(z)^\alpha \ne 0.$ For $f\in M^n$ and given by \[f(z) = z + a_2z^2 + a_3 z^3 + ... ,\] sharp upper bounds are obtained for the Fekete-Szeg\"{o} functional $\big|a_3-\mu a_2^2 \big|$ when $\mu$ is real. Keywords: Convex function; starlike function; Fekete-Szeg\"{o} functional; distortion.