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Mat. Zametki, 1973, Volume 14, Issue 3, Pages 305–316 (Mi mz7260)  

Sharpening certain cyclic inequalities

E. K. Godunova, V. I. Levin

Moscow State Pedagogical Institute

Abstract: This paper studies the lower estimate of cyclic sums of the form
$$\frac1n\sum_{i=1}^n\varphi(\ln\frac{a_{i+1}}{a_i},\ln\frac{a_{i+2}}{a_i+1}),$$
where $\varphi(x,y)$ is a twice continuous differentiable function on the whole plane, $a_{i+n}=a_i$. A structural description is given of a class of functions $\varphi$ for which the lower bound of this sum is attained for $a_i=\mathrm{const}$, i.e., equal to $\varphi(0,0)$. A means of finding the lower bound in all other cases is indicated. This result sharpens and generalizes a number of well known cyclic inequalities.

Full text: PDF file (669 kB)

English version:
Mathematical Notes, 1973, 14:3, 735–741

Bibliographic databases:

UDC: 517.5
Received: 05.06.1972

Citation: E. K. Godunova, V. I. Levin, “Sharpening certain cyclic inequalities”, Mat. Zametki, 14:3 (1973), 305–316; Math. Notes, 14:3 (1973), 735–741

Citation in format AMSBIB
\Bibitem{GodLev73}
\by E.~K.~Godunova, V.~I.~Levin
\paper Sharpening certain cyclic inequalities
\jour Mat. Zametki
\yr 1973
\vol 14
\issue 3
\pages 305--316
\mathnet{http://mi.mathnet.ru/mz7260}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=340522}
\zmath{https://zbmath.org/?q=an:0284.26009}
\transl
\jour Math. Notes
\yr 1973
\vol 14
\issue 3
\pages 735--741
\crossref{https://doi.org/10.1007/BF01147447}


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