A mean-value theorem and its applications

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:373

Article

Matkowski, Janusz^1,2 

[1] Univ Zielona Gora, Fac Math Comp Sci & Econometr, PL-65246 Zielona Gora, Poland

[2] Silesian Univ, Inst Math, PL-42007 Katowice, Poland

关键词: Mean-value theorem;    Lagrange mean-value theorem;    Cauchy mean-value theorem;    Mean;    Stolarsky means;    Invariant mean;    Mean-type mapping;    Iteration;    Functional equation;   

DOI  :  10.1016/j.jmaa.2010.06.057

来源: Elsevier

【 摘 要 】

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean Lill, we prove that there exists a unique two variable mean M[f] such that f (x) - f (y)/x-y = M[f] (f'(x), f' (y)) for all x, y is an element of I, x not equal y. The mean Mill is closely related L[f]. Necessary and sufficient condition for the equality Mill = Migl is given. A family of means [M([t]): t is an element of R) relevant to the logarithmic means is introduced. The invariance of geometric mean with respect to mean-type mappings of this type is considered. A result on convergence of the sequences of iterates of some mean-type mappings and its application in solving some functional equations is given. A counterpart of the Cauchy mean-value theorem is presented. Some relations between Stolarsky means and M([t]) means are discussed. (c) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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