On some new length problem for analytic functions

Yıl 2017, Cilt: 46 Sayı: 3, 427 - 435, 01.06.2017

Janusz Sokó\l{} Mamoru Nunokawa

Öz

Let $\mathcal{H}$ denote the class of analytic functions in the unit disk $|z|<1$. Let $C(r,f)$ be the closed curve which is the image of the circle $|z|=r<1$ under the mapping $w=f(z)\in\mathcal{H}$, $L(r,f)$ the length of $C(r,f)$ and let $A(r,f)$ be the area enclosed by $C(r,f)$. Let $l(re^{i\theta},f)$ be the length of the image curve of the line segment joining $re^{i\theta}$ and $re^{i(\theta+\pi)}$ under the mapping $w=f(z)$ and let $l(r,f)=\max_{0\leq\theta 2 \pi}l(re^{i\theta},f)$. We find upper bound for $l(r,f)$ for $f(z)$ in some known classes of functions. Moreover, we prove that $l(r,f)=\mathcal{O}\left( \log\frac{1}{1-r} \right)$ and that $L(r,f)=\mathcal{O}\left\{ A(r,f)\log \frac{1}{1-r}\right\}^{1/2}$ as $r\to 1$ under weaker assumptions on $f(z)$ than some previous results of this type.

Anahtar Kelimeler

length problem, convex functions, starlike functions

Kaynakça

• P. Eenigenburg, On the radius of curvature for convex analytic functions, Canad. J. Math. 22(3)(1970) 486491.
• L. Fejér, F. Riesz, Über einige funktionentheoretische Ungleichungen, Math. Zeitschr. 11(1921) 305314.
• A. W. Goodman, Univalent Functions, Vols. I and II, Mariner Publishing Co.: Tampa, Florida (1983).
• W. F. Hayman, The asymptotic behaviour of p-valent functions, Proc. London Math. Soc. 3(5)(1955) 257284.
• F. R. Keogh, Some theorems on conformal mapping of bounded star-shaped domain, Proc. London Math. Soc. (3)9(1959) 481491.
• M. Nunokawa, On the Univalency and Multivalency of Certain Analytic Functions, Math. Zeitschr. 104(1968) 394404.
• M. Nunokawa, S. Owa, S. Fukui, H. Saitoh, M.-P. Chen, A class of functions which do not assume non-positive real part, Tamkang J. Math. 19(2)(1968) 2326.
• M. Nunokawa, On Bazilevic and convex functions, Trans. Amer. Math. Soc., 143(1969) 337341.
• M. Nunokawa, A note on convex and Bazilevic functions, Proc. Amer. Math. Soc., 24(2)(1970) 332335.
• M. Nunokawa, J. Sokó\l{}, On some length problems for analytic functions, Osaka J. Math. 51(2014) 695707.
• M. Nunokawa, J. Sokó\l{}, On some length problems for univalent functions, Math. Meth. Appl. Sci., 39(7)(2016) 16621666.
• Ch. Pommerenke, Über nahezu konvexe analytische Functionen, Arch. Math. (Basel) 16(1965) 344347.
• E. Study, Konforme Abbildung Einfachzusammenhangender Bereiche, B. C. Teubner, Leipzig und Berlin 1913.
• D. K. Thomas, On starlike and close-to-convex univalent functions, J. London Math. Soc. 42(1967) 427435.
• D. K. Thomas, A note on starlike functions, J. London Math. Soc. 43(1968) 703706.
• M. Tsuji, Complex Functions Theory, Maki Book Comp., Tokyo 1968.(Japanese)