An estimate of the curvature of the images of circles under maps given by convex univalent functions in a disk (Q1326016)

Let \(S^ 0_ p\) denote the class of functions \(f(z)= z+ \sum^ \infty_{n=1} c_{np+1} z^{np+1}\) holomorphic and convex in the disc \(E= \{z: | z|< 1\}\). In this paper, the sharp estimates for the curvature of the images of the circles \(\delta D_ \rho= \{z: z= z_ 0+ \rho e^{i\varphi},\;0\leq \varphi< 2\pi\}\) at \(w= f(z)\) are found for the class \(S^ 0_ p\), \(p= 1,2,\dots\) . The formula for the curvature \(K(w)\) of the image \(\delta D_ p\) at \(w= f(z)\) has the form \[ K(w)= {\text{Re}\{1+ (z- z_ 0)f''(z)/f'(z)\}\over \rho| f'(z)|}. \] For \(p=1\) we have the following theorem. The curvature \(K(w)\) of the image of \(\delta D_ \rho\) by the function \(f\) of the class \(S^ 0_ 1\) at the point \(w= f(z)\), \(z= r_ 0+\rho= r<1\), \(0< r_ 0< 1\), \(0< \rho< 1-r_ 0\), satisfies the following sharp estimates \[ {1- r^ 2\over r} (b+ 1)\leq K(w)\leq \begin{cases} {1- r^ 2\over r} {a-1\over a\ln a} a^{(a(b+1)/(a- 1)- 1/\ln a)} &\text{if }0< r_ 0\leq d\\ {1-r^ 2\over r} (ab+1) &\text{if }d\leq r_ 0< r,\end{cases} \] where \[ a=\left({1+ r\over 1- r}\right)^ 2,\quad b= {r_ 0(1-r)\over (r- r_ 0)(1+ r)},\quad d= {r(1+ r)(a- 1-\ln a)\over (a-1)(1+ r)+ \ln a[a(1- r)- 1-r]}. \] For arbitrary \(p\) the formula is too complicated to be presented here. In the proof the result of \textit{I. A. Aleksandrov} and \textit{V. V. Chernikov} [Sib. Mat. Zh. 4, 1201-1207 (1963; Zbl 0138.064)] is applied. By this result it is enough to find the extremum of curvature only for functions \(f(z)\) such that \(zf'(z)= g(z)\), where \[ g(z)= z(1- z^ p e^{-it_ 1})^{-2(1-\lambda)/p} (1- z^ p e^{-it_ 2})^{- 2\lambda/p}, \] \(0< \lambda< 1\),\(-\pi\leq t_ 1\leq t_ 2\leq \pi\).