For an analytic function $f$ defined on the unit disk $|z|<1$, let $Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<rle 1$. In 1990, Yamashita conjectured that $Delta(r,z/f)le pi r^2$ for convex functions $f$ and i
t was finally settled in 2013 by Obradovi{c} and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation $zf(z)/f(z)prec (1+(1-2beta)alpha z)/(1-alpha z)$ for $0le beta<1$ and $0<alphale 1$. We prove Yamashitas conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.
A motivation comes from {em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference operator acti
ng on analytic functions in the unit disk $mathbb{D}={zin mathbb{C}:,|z|<1}$. We use the terminology {em $q$-close-to-convex functions} for the $q$-analog of close-to-convex functions. The $q$-theory has wide applications in special functions and quantum physics which makes the study interesting and pertinent in this field. In this paper, we obtain some interesting results concerning conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in the $q$-close-to-convex family. As a result we find certain dilogarithm functions that are contained in this family. Secondly, we also study the famous Bieberbach conjecture problem on coefficients of analytic $q$-close-to-convex functions. This produces several power series of analytic functions convergent to basic hypergeometric functions.
