## 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

### MSC:

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

Zbl 0634.30011
Full Text:

### References:

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