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Journal: Annales Mathematicae Silesianae

New and original integral inequalities under monotonicity and convexity assumptions

Christophe Chesneau

Language: EN | Published: 02-07-2025 | Abstract

This article examines integral inequalities dealing with functions of the form “a function raised to the power of another function” under varying monotonicity and convexity assumptions. First, we assess the validity of a referenced theorem on the subject. Specifically, we present a counterexample and identify a gap in its proof. We then propose an alternative version of the theorem with more flexible convexity assumptions. In addition, we establish new lower and upper bounds for the same integral using refined Hermite–Hadamard integral inequalities. A complementary variant is also discussed. Thus, our results fill gaps in the literature and extend existing results on integral inequalities under classical assumptions.

2025-07-02



Journal: Annales Mathematicae Silesianae

Inequalities of Lipschitz type for power series in Banach algebras

Sever S. Dragomir

Language: EN | Published: 30-09-2015 | Abstract | pp. 61-83

Let f(z) = Σ_n=0^∞α_nzⁿ be a function defined by power series with complex coefficients and convergent on the open disk D(0,R)⊂ℂ, R>0. For any x,y∈????, a Banach algebra, with ‖x‖,‖y‖<R we show among others that
‖f(y)-f(x)‖ ≤ ‖y-x‖∫₀¹f'_a(‖(1-t)x+ty‖)dt
where f_a(z) = Σ_n=0^∞|α_n|zⁿ. Inequalities for the commutator such as
‖f(x)f(y) - f(y)f(x)‖ ≤ 2f_a(M)f'_a(M)‖y-x‖,
if ‖x‖,‖y‖≤M<R, as well as some inequalities of Hermite–Hadamard type are also provided.

2015-09-30