Geodesic
Radii of convexity and close-to-convexity of certain integral representations
Matematičeskie zametki, Tome 7 (1970) no. 5, pp. 581-592 
F. G. Avkhadiev. Radii of convexity and close-to-convexity of certain integral representations. Matematičeskie zametki, Tome 7 (1970) no. 5, pp. 581-592. http://geodesic.mathdoc.fr/item/MZM_1970_7_5_a6/
@article{MZM_1970_7_5_a6,
     author = {F. G. Avkhadiev},
     title = {Radii of convexity and close-to-convexity of certain integral representations},
     journal = {Matemati\v{c}eskie zametki},
     pages = {581--592},
     year = {1970},
     volume = {7},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_7_5_a6/}
}
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%G ru
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Voir la notice de l'article provenant de la source Math-Net.Ru

Strict upper bounds are determined for $|s(z)|$, $|\mathrm{Re}\,s(z)|$, and $|\mathrm{Im}\,s(z)|$ in the class of functions $s(z)=a_nz^n+a_{n+1}z^{n+1}+\dots$ ($n\geqslant1$) regular in $|z|<1$ and satisfying the condition $$ |u(\theta_1)-u(\theta_2)|\leqslant K|\theta_1-\theta_2|, $$ where $u(\theta)=\mathrm{Re}\,s(e^{i\theta})$, $K>0$, and $\theta_1$ and $\theta_2$ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.