Estimates of the \(n\)th divided difference. (English. Russian original) Zbl 0692.30009

Lith. Math. J. 29, No. 3, 255-265 (1989); translation from Lit. Mat. Sb. 29, No. 3, 491-506 (1989).
Let \(\Phi (z)\) be a function defined in the unit disc \(D\) such that \(\Phi^{(n)}(z)\) is continuous in \(D\). For given points \(z_ k\), \(0\leq k\leq n\), the \(n\)th divided difference is defined by the formula \[ [\Phi (z);z_ 0,z_ 1,...,z_ n]=\int^{1}_{0}\int^{t_ 1}_{0}\dots\int^{t_{n-1}}_{0}\Phi^{(n)}(u)\,dt_ 1\dots\,dt_ n, \] where \(u=z_ 0+(z_ 1-z_ 0)t_ 1+\dots +(z_ n-z_{n-1})t_ n\) and \(0\leq t_ 1\leq 1\), \(0\leq t_ 2\leq t_ 1,\dots\), \(0\leq t_ n\leq t_{n- 1}\). The present paper being a continuation of earlier studies by the present author and J. B. Mieliauskas [Litov. Mat. Sb. 27, 273–278 (1987; Zbl 0634.30011)] brings upper estimates of the \(n\)th divided difference and its real part. Results have the following form:
If \(| \Phi^{(n)}(z)| \leq M^{(n)}(| z|)\) then \[ | [\Phi (z);z_ 0,z_ 1,\dots,z_ n|]\leq [M(| z|);| z|,| z_ 1|,\dots,| z_ n|] \] and are too long to be quoted here.
Reviewer: E.Złotkiewicz


30C35 General theory of conformal mappings
30C75 Extremal problems for conformal and quasiconformal mappings, other methods


Zbl 0634.30011
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