## Functional inequalities associated with Jordan-von Neumann-type additive functional equations.(English)Zbl 1133.39024

Summary: We prove the generalized Hyers-Ulam stability of the following functional inequalities:
\begin{aligned} \| f(x)+f(y)+f(z)\| &\leq \| 2f((x+y+z)/2)\|, \\ \| f(x)+f(y)+f(z)\| &\leq \| f(x+y+z)\|, \\ \| f(x)+f(y)+2f(z)\|&\leq \| 2f((x+y)/2+z)\| \end{aligned}
in the spirit of the Rassias stability approach for approximately homomorphisms.

### MSC:

 39B62 Functional inequalities, including subadditivity, convexity, etc. 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
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### References:

 [1] Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150. [2] Hyers, DH, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224, (1941) · Zbl 0061.26403 [3] Rassias, ThM, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 297-300, (1978) · Zbl 0398.47040 [4] Rassias, ThM, Problem 16; 2, report of the 27th international symp. on functional equations, Aequationes Mathematicae, 39, 292-293; 309, (1990) [5] Gajda, Z, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, 14, 431-434, (1991) · Zbl 0739.39013 [6] Rassias, ThM; Šemrl, P, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proceedings of the American Mathematical Society, 114, 989-993, (1992) · Zbl 0761.47004 [7] Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410. · Zbl 1011.39019 [8] Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser Boston, Boston, Mass, USA; 1998:vi+313. [9] Rassias, JM, On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis, 46, 126-130, (1982) · Zbl 0482.47033 [10] Găvruţa, P, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184, 431-436, (1994) · Zbl 0818.46043 [11] Jun, K-W; Lee, Y-H, A generalization of the Hyers-Ulam-Rassias stability of the pexiderized quadratic equations, Journal of Mathematical Analysis and Applications, 297, 70-86, (2004) · Zbl 1060.39029 [12] Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256. · Zbl 0980.39024 [13] Park, C, Homomorphisms between poisson[inlineequation not available: see fulltext.]-algebras, Bulletin of the Brazilian Mathematical Society. New Series, 36, 79-97, (2005) · Zbl 1091.39007 [14] Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. to appear in Bulletin des Sciences Mathématiques to appear in Bulletin des Sciences Mathématiques · Zbl 0761.47004 [15] Gilányi, A, Eine zur parallelogrammgleichung äquivalente ungleichung, Aequationes Mathematicae, 62, 303-309, (2001) · Zbl 0992.39026 [16] Rätz, J, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Mathematicae, 66, 191-200, (2003) · Zbl 1078.39026 [17] Gilányi, A, On a problem by K. nikodem, Mathematical Inequalities & Applications, 5, 707-710, (2002) · Zbl 1036.39020 [18] Fechner, W, Stability of a functional inequality associated with the Jordan-von Neumann functional equation, Aequationes Mathematicae, 71, 149-161, (2006) · Zbl 1098.39019
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