On the special form of integral convolution type inequality due to Walter and Weckesser

Walter and Weckesser’s result (Aequationes Math 46:212–219, 1993), extending the Bushell–Okrasiński convolution type inequality (Bushell and Okrasiński in J Lond Math Soc (2) 41:503–510, 1990), gave some general conditions on the functions k:0,d→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k:\left[ 0,d\right) \rightarrow \mathbb {R}$$\end{document} and g:0,∞→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:\left[ 0,\infty \right) \rightarrow \mathbb {R}$$\end{document} under which, for every increasing function f : 0,d→0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ 0,d\right) \rightarrow \left[ 0,\infty \right) $$\end{document}, the inequality ∫0xkx-sgfsds≤g∫0xfsds,x∈0,d,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{0}^{x}k\left( x-s\right) g\left( f\left( s\right) \right) ds\le g\left( \int _{0}^{x}f\left( s\right) ds\right) ,\quad x\in \left( 0,d\right) , \end{aligned}$$\end{document}is satisfied. Applying the result on a simultaneous system of functional inequalities, we prove that if d>1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>1,$$\end{document} then, in general, both k and g must be power functions.


Introduction
where d ∈ (0, 1], p ≥ 1 and f : [0, d] → [0, ∞) is an arbitrary continuous increasing (throughout the paper we use terms increasing and decreasing in the weak sense) function, is called the classical Bushell-Okrasiński inequality (the non-classical forms of (1.1) include its generalizations to various kinds of integrals-for instance, fuzzy integrals like the Sugeno integral [8] or the universal integral [1]). It was proved in [3] as an auxiliary result in the study of the existence of solutions of some class of Volterra integral equations and almost immediately questions about an extension of (1.1) arose [9]. The first such extension was given by Walter and Weckesser in [10], but also later many other papers were published, cointaing results that in particular lead to the Bushell-Okrasiński inequality (see [4,5,7]).
In the aforementioned article [10] Walter and Weckesser proved the following theorem: Theorem 1. Let for every c ∈ (0, d] the function h c be defined by If one of the following two conditions is satisfied: It is easy to observe that the Bushell-Okrasiński inequality (1.1) can be obtained from Theorem 1 by taking functions g and k, both of which are power functions with the exponent p ≥ 1 (it is worth noting that now the assumption d ≤ 1 is no longer needed). Because of the importance of the Bushell-Okrasiński inequality, the natural question arises: under what additional conditions does an inequality in the Walter-Weckesser theorem reduce to the Bushell-Okrasiński inequality? In this paper we give a partial answer to this question. We show that such a reduction is enforced if, in essential, d > 1 and functions g and K are positive, provided that K also satisfies a certain inequality. The main tool that we use to prove this is a certain result on the solutions of the simultaneous system of functional inequalities stated in the next section as Theorem 2.

Auxiliary results
where Q denotes the set of rational numbers. Suppose that a function g : (0, ∞) → R satisfies the system of inequalities g is continuous at least at one point, and g ((0, ∞)) (−∞, 0). Then either The suitable result for the reversed inequalities reads as follows.
Suppose that a function g : (0, ∞) → R satisfies the system of inequalities g is continuous at least at one point, and g ((0, ∞)) (−∞, 0). Then either Let d > 0 be arbitrarily fixed. In order to directly use Theorems 2 and 3, we must relax some of the assumptions about the domains g and K made in Theorem 1, so in this section let g : and let a bivariable function h : (0, d)×(0, ∞) → R be defined by the following formula Applying Theorems 2 and 3, we show now that the constant sign of h implies that g must be a power function. Namely, we have the following If 1 < d ≤ ∞, and one of the following two conditions is satisfied: Proof. Assume that (i) holds true. Putting and using the nonnegativity of h, we notice that g satisfies the following system of inequalities Because g is continuous at a point, Theorem 3 implies the result. In case (ii), applying Theorem 2, we argue analogously.
This theorem does not give any specific information about K. It turns out, however, that if g is positive and K satisfies an additional condition, which is quite natural in the context of the inequality log K(b) log a in (i) or its converse in (ii), then both g and K must be power functions of the same exponent. Namely, we have the following If 1 < d ≤ ∞, and one of the following two conditions is satisfied: (i) the function h is nonnegative and (ii) the function h is non positive and then either g (y) = 0, y > 0, or g is positive, for some real p g (y) = g (1) y p for all y ∈ (0, ∞) ; and h is equal to zero, i.e., Proof. Assume that condition (i) holds true. For an arbitrarily fixed number a ∈ (0, 1), choose b ∈ (1, d) such that log a and log b are incommensurable and put α := K (a) , The assumed nonnegativity of h implies that g satisfies the simultaneous system of inequalities g (ay) ≥ αg (y) , g (by) ≥ βg (y) , y > 0.
Since g is continuous at a point, in view of Theorem 3, either g ≡ 0 or To find the form of K in the latter case, notice that from the definitions of α, β and p, we have Since the definition of p does not depend on b, and the set of the numbers b such that log b log a / ∈ Q is dense in (1, d), the continuity of K implies that K(x) = x p for 14 T. Ma lolepszy, J. Matkowski AEM all x ∈ [1, d). Interchanging the roles of a and b in this reasoning we conclude that K (x) = x p for all x ∈ (0, 1). Thus we have shown that Now, for all x ∈ (0, d) and y > 0, making use of the definition of h, we get which shows that In case (ii), the simultaneous system of inequalities is reversed, so applying Theorem 2, we can argue similarly. This completes the proof. Remark 1. In Theorem 5 condition (i), which is equivalent to the implication: for all s, t, 1/t is increasing. Similarly, condition (ii), which is equivalent to the implication: for all s, t, 1/t is decreasing.

Main results
From now on, following [10], we assume that functions g and K are defined on larger intervals, i.e. on [0, ∞) and (0, d], respectively. Based on Theorems 4 and 5, we obtain certain refinements of Theorem 1. To be more precise, with some additional assumptions about K (the same ones as in Theorems 4 and 5, respectively) we find all admissible forms of the nonnegative function g in Theorem 1, provided that d > 1. (i) If there exist a, b ∈ (0, d] such that or g is positive on (0, ∞) and Proof. (i) The function g, being a convex function, is in particular continuous in (0, ∞). The nonnegativity of h c and Theorem 4 imply that either g is zero in (0, ∞) or g (y) = g(1)y p for all y > 0, with log b ≥ 1. Moreover, the extension of g to the interval [0, ∞) by putting g(0) := α, α ≥ 0, preserves its convexity. To finish the proof, we have to show that such extensions preserve the nonnegativity and the increasingness of h c . When g ≡ 0 on (0, ∞), then, for all c ∈ (0, d], and h c (y) = 0, y > 0, hence α = 0, as h c (0) ≤ h c (y) for all y > 0. We obtain the same value of α for the second type of function g, i.e. g (y) = g(1)y p for y ∈ (0, ∞). In this case we have and h c (y) = g(1)y p (c p − K(c)), c ∈ (0, d], y > 0. As lim y→0 + h c (y) = 0, the increasingness of h c implies that α = 0. (ii) In this case a similar argument works. The difference in the thesis comes from the fact that the convexity of power functions fails when p ∈ (0, 1).
A similar argument in combination with Theorem 5 gives and for every c ∈ (0, d] let the function h c be nonnegative and increasing.

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T. Ma lolepszy, J. Matkowski AEM or g is positive on (0, ∞) and for some real p > 1 g (y) = g (1) y p for all y ∈ (0, ∞) , and If g is concave, the corresponding results read as follows: , and for every c ∈ (0, d] let the function h c be nonnegative and decreasing. or g is positive on (0, ∞) and and for every c ∈ (0, d] let the function h c be nonnegative and decreasing. then either g ≡ 0, or g is positive on (0, ∞) and for some real p < 1 g (y) = g (1) y p for all y ∈ (0, ∞) , and An immediate consequence of Corollary 1 is the following useful result showing that if in the Walter-Weckesser theorem d > 1 and k is taken in such a way that K is a power function, then the number of possible convex functions g is quite limited. = p, so the application of Corollary 1 ends the proof.