ON OZAKI CLOSE-TO-CONVEX FUNCTIONS
Journal
Bulletin of the Australian Mathematical Society
Date Issued
2018-09-20
Author(s)
ALLU, VASUDEVARAO
THOMAS, DEREK K.
TUNESKI, NIKOLA
DOI
10.1017/s0004972718000989
Abstract
<jats:p>Let <jats:inline-formula>
<jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972718000989_inline1" xlink:type="simple" /><jats:tex-math>$f$</jats:tex-math></jats:alternatives>
</jats:inline-formula> be analytic in <jats:inline-formula>
<jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972718000989_inline2" xlink:type="simple" /><jats:tex-math>$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$</jats:tex-math></jats:alternatives>
</jats:inline-formula> and given by <jats:inline-formula>
<jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972718000989_inline3" xlink:type="simple" /><jats:tex-math>$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$</jats:tex-math></jats:alternatives>
</jats:inline-formula>. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.</jats:p>
<jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972718000989_inline1" xlink:type="simple" /><jats:tex-math>$f$</jats:tex-math></jats:alternatives>
</jats:inline-formula> be analytic in <jats:inline-formula>
<jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972718000989_inline2" xlink:type="simple" /><jats:tex-math>$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$</jats:tex-math></jats:alternatives>
</jats:inline-formula> and given by <jats:inline-formula>
<jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972718000989_inline3" xlink:type="simple" /><jats:tex-math>$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$</jats:tex-math></jats:alternatives>
</jats:inline-formula>. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.</jats:p>