The concept of l-gap convex functions is defined, which is more general than convex functions and allows some non-convex parts of the function. A Jensen type inequality is established and some examples are discussed. As an application and generalization, we prove that the majorization theorem also holds for l-gap convex functions. Then we use the conclusions from the above sections to establish a Hermite-Hadamard type inequality for l-gap convex functions.
In this paper, we use (m+4)-convex functions to derive an estimate for Jensen’s inequality in the context of divided differences. In addition, we extend these results for (h,g;α−n)-convex functions. Finally, we present some results for g-convex functions, (h,g)-convex functions and provide a discussion and examples concerning h-convex functions.
Language:
EN
| Published:
30-09-1992
|
Abstract
| pp. 30-41
There is defined quasi-Jensen function as a solution of a certain functional inequality which generalizes the classical Jensen equation: f((x+y)/2) = (f(x)+f(y))/2. The introduced inequality is analogous to the inequality which defines J. Tabor's quasi-additive functions. The main result of this paper is to show strong relationship between quasi-Jensen and quasi-additive functions.
Language:
EN
| Published:
02-07-2025
|
Abstract
| pp. 42-58
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.
Teodoro Lara
,
Nelson Merentes
,
Edgar Rosales
,
Ambrosio Tineo
Language:
EN
| Published:
31-01-2018
|
Abstract
| pp. 237-245
In this research we deal with algebraic properties and characterizations of convex functions in the context of a time scale; this notion of convexity has been studied for some other authors but the setting of properties are establish here. Moreover, characterizations, a separation theorem and an inequality of Jensen type for this class of functions are shown as well.
2018-01-31
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English