• Title, Summary, Keyword: Jensen type function

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ON THE STABILITY OF A JENSEN TYPE FUNCTIONAL EQUATION ON GROUPS

  • FAIZIEV VALERH A.;SAHOO PRASANNA K.
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.757-776
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    • 2005
  • In this paper we establish the stability of a Jensen type functional equation, namely f(xy) - f($xy^{-1}$) = 2f(y), on some classes of groups. We prove that any group A can be embedded into some group G such that the Jensen type functional equation is stable on G. We also prove that the Jensen type functional equation is stable on any metabelian group, GL(n, $\mathbb{C}$), SL(n, $\mathbb{C}$), and T(n, $\mathbb{C}$).

SUPERQUADRATIC FUNCTIONS AND REFINEMENTS OF SOME CLASSICAL INEQUALITIES

  • Banic, Senka;Pecaric, Josip;Varosanec, Sanja
    • Journal of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.513-525
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    • 2008
  • Using known properties of superquadratic functions we obtain a sequence of inequalities for superquadratic functions such as the Converse and the Reverse Jensen type inequalities, the Giaccardi and the Petrovic type inequalities and Hermite-Hadamard's inequalities. Especially, when the superquadratic function is convex at the same time, then we get refinements of classical known results for convex functions. Some other properties of superquadratic functions are also given.

APPROXIMATELY ADDITIVE MAPPINGS IN NON-ARCHIMEDEAN NORMED SPACES

  • Mirmostafaee, Alireza Kamel
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.2
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    • pp.387-400
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    • 2009
  • We establish a new strategy to study the Hyers-Ulam-Rassias stability of the Cauchy and Jensen equations in non-Archimedean normed spaces. We will also show that under some restrictions, every function which satisfies certain inequalities can be approximated by an additive mapping in non-Archimedean normed spaces. Some applications of our results will be exhibited. In particular, we will see that some results about stability and additive mappings in real normed spaces are not valid in non-Archimedean normed spaces.

ON THE SUPERSTABILITY OF SOME PEXIDER TYPE FUNCTIONAL EQUATION II

  • Kim, Gwang-Hui
    • The Pure and Applied Mathematics
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    • v.17 no.4
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    • pp.397-411
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    • 2010
  • In this paper, we will investigate the superstability for the sine functional equation from the following Pexider type functional equation: $f(x+y)-g(x-y)={\lambda}{\cdot}h(x)k(y)$ ${\lambda}$: constant, which can be considered an exponential type functional equation, the mixed functional equation of the trigonometric function, the mixed functional equation of the hyperbolic function, and the Jensen type equation.

JORDAN-VON NEUMANN TYPE FUNCTIONAL INEQUALITIES

  • Kwon, Young Hak;Lee, Ho Min;Sim, Jeong Soo;Yang, Jeha;Park, Choonkil
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.3
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    • pp.269-277
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    • 2007
  • It is shown that $f:\mathbb{R}{\rightarrow}\mathbb{R}$ satisfies the following functional inequalities (0.1) ${\mid}f(x)+f(y){\mid}{\leq}{\mid}f(x+y){\mid}$, (0.2) ${\mid}f(x)+f(y){\mid}{\leq}{\mid}2f(\frac{x+y}{2}){\mid}$, (0.3) ${\mid}f(x)+f(y)-2f(\frac{x-y}{2}){\mid}{\leq}{\mid}2f(\frac{x+y}{2}){\mid}$, respectively, then the function $f:\mathbb{R}{\rightarrow}\mathbb{R}$ satisfies the Cauchy functional equation, the Jensen functional equation and the Jensen quadratic functional equation, respectively.

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APPROXIMATION OF CAUCHY ADDITIVE MAPPINGS

  • Roh, Jai-Ok;Shin, Hui-Joung
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.851-860
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    • 2007
  • In this paper, we prove that a function satisfying the following inequality $${\parallel}f(x)+2f(y)+2f(z){\parallel}{\leq}{\parallel}2f(\frac{x}{2}+y+z){\parallel}+{\epsilon}({\parallel}x{\parallel}^r{\cdot}{\parallel}y{\parallel}^r{\cdot}{\parallel}z{\parallel}^r)$$ for all x, y, z ${\in}$ X and for $\epsilon{\geq}0$, is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.

UNIFORMLY BOUNDED COMPOSITION OPERATORS ON A BANACH SPACE OF BOUNDED WIENER-YOUNG VARIATION FUNCTIONS

  • Glazowska, Dorota;Guerrero, Jose Atilio;Matkowski, Janusz;Merentes, Nelson
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.675-685
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    • 2013
  • We prove, under some general assumptions, that a generator of any uniformly bounded Nemytskij operator, mapping a subset of space of functions of bounded variation in the sense of Wiener-Young into another space of this type, must be an affine function with respect to the second variable.