A characterization of the sine function by functional inequalities

Tyrala, Iwona

doi:10.7153/mia-14-02

Mathematical Inequalities &Amp; Applications

2011

DOI: 10.7153/mia-14-02

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A characterization of the sine function by functional inequalities

Iwona Tyrala¹

Abstract: In the present paper we deal with the first generalization of Wilson's differenceassuming that its absolute value is majorized by some function in a single variable.Mathematics subject classification (2010): 39B62, 39B82.

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2024

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Some Comments on the Superstability of a General Functional Equation

Brzdęk

2024

Symmetry

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In this paper, we prove a superstability theorem for a general functional equation ∑j=1∞ajf(γj(t,s))=h(t)g(s), with the unknown functions g:T→X, h:S→K and f:S→X, such that the series ∑j=1∞f(γj(t,s)) is convergent for every (s,t)∈S×T, where S and T are nonempty sets, and X is a Banach space over a field K, which is either the set of real numbers R or the set of complex numbers C. Namely, we show that if h is unbounded, and the difference ∑j=1∞ajf(γj(t,s))−h(t)g(s) is bounded, then h and g satisfy the equation ∑j=1∞ajg(γj(t,s))=h(t)g(s). Next, we show that the superstability of pexiderizations and radical versions of several well-known functional equations (e.g., of Cauchy, d’Alembert, Wilson, Reynolds, and homogeneity) is a consequence of this simple outcome. In this way, we generalize several classical superstability results and, in particular, the first superstability outcome of J. Baker, J. Lawrence, and F. Zorzitto, which states that every unbounded mapping f, from a real vector space X into the set of real numbers R, satisfying the inequality |f(x+y)−f(x)f(y)|≤δ for every x,y∈X, with some real δ>0, is an exponential function, i.e., satisfies the equality f(x+y)=f(x)f(y) for all x,y∈X. In order to make this publication more accessible to a wider range of readers, we limit various related information, avoid very abstract generalizations, and provide some simple examples.

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Some Comments on the Superstability of a General Functional Equation

Brzdęk

2024

Symmetry

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A characterization of the sine function by functional inequalities

Abstract: In the present paper we deal with the first generalization of Wilson's differenceassuming that its absolute value is majorized by some function in a single variable.Mathematics subject classification (2010): 39B62, 39B82.

Cited by 1 publication

References 9 publications

Some Comments on the Superstability of a General Functional Equation

Some Comments on the Superstability of a General Functional Equation

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