## The punishing factors for convex pairs are $$2^{n-1}$$.(English)Zbl 1147.30014

Let $$\Omega$$ and $$\Pi$$ be simply connected proper subdomains of the complex plane, $$f:\Omega\rightarrow\Pi$$ be a holomorphic function, $$\lambda_\Omega(z)$$ and $$\lambda_\Pi(z)$$ be the density of the Poincaré metric with curvature $$K=-4$$ at $$z\in\Omega$$ and $$w\in\Pi,$$ respectively.
The authors prove that if $$\Omega$$ and $$\Pi$$ are convex then $\frac{|f^{(n)}(z)|}{n!}\leq 2^{n-1}\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))}$ and if $$\Omega$$ is convex and $$\Pi$$ linearly accessible then $\frac{|f^{(n)}(z)|}{(n+1)!}\leq 2^{n-2}\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))}$ for $$n=2,3,\ldots$$ and $$z\in\Omega.$$ Even more $$2^{n-1}$$ and $$2^{n-2}$$ are the best possible for any admissible pair $$(\Omega,\Pi).$$
The title of the paper is motivated by the fact that the quotient $$\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))}$$ reflects the influence of the position of the points $$z$$ and $$f(z)$$ in $$\Omega$$ and $$\Pi$$ on the $$f^{(n)}(z)$$ and the coefficients $$2^{n-1}$$ and $$2^{n-2}$$ are factors that “punish” bad behavior of $$\Omega$$ and $$\Pi$$ at the boundary.

### MSC:

 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30D50 Blaschke products, etc. (MSC2000)
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### References:

 [1] Ahlfors, L. V.: Conformal invariants: topics in geometric function theory . McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973. · Zbl 0272.30012 [2] Avkhadiev, F. G. and Wirths, K.-J.: Schwarz-Pick inequalities for derivatives of arbitrary order. Constr. Approx. 19 (2003), 265-277. · Zbl 1018.30018 [3] Avkhadiev, F. G. and Wirths, K.-J.: Punishing factors for angles. Comput. Methods Funct. Theory 3 (2003), 127-141. · Zbl 1072.30014 [4] Avkhadiev, F. G. and Wirths, K.-J.: Schwarz-Pick inequalities for hyperbolic domains in the extended plane. Geom. Dedicata 106 (2004), 1-10. · Zbl 1066.30025 [5] Avkhadiev, F. G. and Wirths, K.-J.: Sharp bounds for sums of coefficients of inverses of convex functions. Comput. Methods Funct. Theory 7 (2007), 105-109. · Zbl 1147.30013 [6] Avkhadiev, F. G. and Wirths, K.-J.: Punishing factors and Chua’s conjecture. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 333-340. · Zbl 1129.30013 [7] Avkhadiev, F. G., Pommerenke, Ch. and Wirths, K.-J.: Sharp inequalities for the coefficients of concave schlicht functions. Comment. Math. Helv. 81 (2006), 801-807. · Zbl 1210.30005 [8] Biernacki, M.: Sur la représentation conforme des domaines linéarement accessibles. Prace Mat. Fiz. 44 (1936), 293-314. · Zbl 0014.02405 [9] Brickman, L.: Subordinate families of analytic functions. Illinois J. Math. 15 (1971), 241-248. · Zbl 0212.10401 [10] Brannan, D. A., Clunie, J. G. and Kirwan, W. E.: On the coefficient problem for functions of bounded boundary rotation. Ann. Acad. Sci. Fenn. Ser. A I 523 (1973). · Zbl 0257.30011 [11] Fejér, L.: Über gewisse durch die Fouriersche und Laplacesche Reihe definierten Mittelkurven und Mittelflächen. Palermo Rend. 38 (1914), 79-97. · JFM 45.0412.03 [12] Hallenbeck, D. J. and MacGregor, T. H.: Linear problems and convexity techniques in geometric function theory . Monographs and Studies in Mathematics 22 . Pitman (Adv. Publishing Program), Boston, MA, 1984. · Zbl 0581.30001 [13] Henrici, P.: Applied and computational complex analysis. Volume 1: Power series-integration-conformal mapping-location of zeros . Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974. · Zbl 0313.30001 [14] Koepf, W.: On close-to-convex functions and linearly accessible domains. Complex Variables Theory Appl. 11 (1989), 269-279. · Zbl 0679.30007 [15] Landau, E.: Einige Bemerkungen über schlichte Abbildung. Jber. Deutsche Math. Verein. 34 (1925/26), 239-243. · JFM 52.0349.03 [16] Marx, A.: Untersuchungen über schlichte Abbildungen. Math. Ann. 107 (1933), 40-67. · Zbl 0005.10901 [17] Miller, S. S. and Mocanu, P. T.: Differential subordinations. Theory and applications . Monographs and Textbooks in Pure and Applied Mathematics 225 . Marcel Dekker, New York, 2000. · Zbl 0954.34003 [18] Pommerenke, Ch.: Univalent functions . Mathematische Lehrbücher 25 . Vandenhoeck & Ruprecht, Göttingen, 1975. · Zbl 0283.30034 [19] Pommerenke, Ch.: Personal Communication, 2002. [20] Ruscheweyh, St.: Über einige Klassen in Einheitskreis holomorpher Funktionen. Ber. Math.-Stat. Sektion Forschungszentrum Graz 7 (1974), 1-12. · Zbl 0274.30006 [21] Ruscheweyh, St.: Two remarks on bounded analytic functions. Serdica 11 (1985), 200-202. · Zbl 0581.30009 [22] Ruscheweyh, St. and Sheil-Small, T.: Hadamard products of schlicht functions and the Pólya-Schoenberg conjecture. Comment. Math. Helv. 48 (1973), 119-135. · Zbl 0261.30015 [23] Schober, G.: Univalent functions-selected topics . Lecture Notes in Mathematics 478 . Springer-Verlag, Berlin-New York, 1975. · Zbl 0306.30018 [24] Strohhäcker, E.: Beiträge zur Theorie der schlichten Funktionen. Math. Z. 37 (1933), 356-380. · JFM 59.0353.02 [25] Suffridge, T. J.: Some remarks on convex maps of the unit disk. Duke Math. J. 37 (1970), 775-777. · Zbl 0206.36202 [26] Szász, O.: Ungleichungen für die Koeffizienten einer Potenzreihe. Math. Z. 1 (1918), 163-183. · JFM 46.0479.01
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