Rocky Mountain Journal of Mathematics

Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras

Choonkil Park, Yuanfeng Jin, and Xiaohong Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we solve the following bi-additive $s$-functional inequalities: $$ \displaylines { \, \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \hfill \cr \hfill \quad \le \Bigl \|s \Bigl (2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) + 2f\Bigl (\frac {x-y}{2}, z+w\Bigr ) - 2f(x,z )+ 2 f(y, w)\Bigr )\Bigr \| ,\! \cr \, \Bigl \|2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) +2 f\Bigl (\frac {x-y}{2}, z+w\Bigr ) -2 f(x,z )+2 f(y, w)\Bigr \| \hfill \cr \hfill \le \|s ( f(x+y, z-w) + f(x-y, z+w) -2f(x,z) +2 f(y, w) )\|, \cr \, } $$ where $s$ is a fixed nonzero complex number with $|s |\lt 1$. We also prove the Hyers-Ulam stability of quasi-multipliers on Banach algebras and unital $C^*$-algebras associated with the bi-additive $s$-functional inequalities above.


The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology, grant No. NRF-2017R1D1A1B04032937. The second author was supported by the National Natural Science Foundation of China, grant No. 11761074.

Article information

Rocky Mountain J. Math., Volume 49, Number 2 (2019), 593-607.

Received: 3 March 2018
Revised: 10 August 2018
First available in Project Euclid: 23 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39B52: Equations for functions with more general domains and/or ranges 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 46L05: General theory of $C^*$-algebras

quasi-multiplier on $C^*$-algebras quasi-multiplier on Banach algebras Hyers-Ulam stability bi-additive $s$-functional inequality


Park, Choonkil; Jin, Yuanfeng; Zhang, Xiaohong. Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras. Rocky Mountain J. Math. 49 (2019), no. 2, 593--607. doi:10.1216/RMJ-2019-49-2-593.

Export citation


  • C.A. Akemann and G.K. Pedersen, Complications of semicontinuity in $C^*$-algebra theory, Duke Math. J. 40 (1973), 785–795.
  • T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.
  • J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in $C^*$-ternary algebras, Bull. Korean Math. Soc. 47 (2010), 195–209.
  • M. Eshaghi Gordji, A. Fazeli and C. Park, $3$-Lie multipliers on Banach $3$-Lie algebras, Int. J. Geom. Meth. Mod. Phys. 9 (2012), article ID 1250052.
  • M. Eshaghi Gordji, M.B. Ghaemi and B. Alizadeh, A fixed point method for perturbation of higher ring derivationsin non-Archimedean Banach algebras, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1611–1625.
  • M. Eshaghi Gordji and N. Ghobadipour, Stability of $(\alpha,\beta,\gamma)$-derivations on Lie $C^*$-algebras, Int. J. Geom. Meth. Mod. Phys. 7 (2010), 1097–1102.
  • W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequat. Math. 71 (2006), 149–161.
  • M. Fošner and J. Vukman, On some functional equations arising from $(m,n)$-Jordan derivations and commutativity of prime rings, Rocky Mountain J. Math. 42 (2012), 1153–1168.
  • P. G\v avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
  • A. Gilányi, Eine zur Parallelogrammgleichung äquivalente Ungleichung, Aequat. Math. 62 (2001), 303–309.
  • ––––, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710.
  • D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), 222–224.
  • R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras: Elementary theory, Academic Press, New York, 1983.
  • J. Lee, A functional equation and degenerate principal series, Rocky Mountain J. Math. 46 (2016), 1987–2016.
  • M. McKennon, Quasi-multipliers, Trans. Amer. Math. Soc. 233 (1977), 105–123.
  • C. Park, Homomorphisms between Poisson $JC^*$-algebras, Bull. Brazilian Math. Soc. 36 (2005), 79–97.
  • C. Park, Additive $\rho$-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26.
  • ––––, Additive $\rho$-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407.
  • Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
  • J. Rätz, On inequalities associated with the Jordan-von Neumann functional equation, Aequat. Math. 66 (2003), 191–200.
  • S.M. Ulam, A collection of the mathematical problems, Interscience Publications, New York, 1960.
  • Z. Wang, Th.M. Rassias and M. Eshaghi Gordji, Stability of quadratic functional equations Šerstnev probabilistic normed spaces, Polit. Univ. Bucharest Sci. Bull. 77 (2015), 79–92.