On Some Generalization of Inequalities of Opial, Yang and Shum

Abstract

In 1960, Z. Opial [20] proved the following interesting integral inequality:Theorem A. If u is a continuously differentiable function on [0, b], and if u(0) = u(b) = 0, and u(x)>0 for x∊(0, b), then1where the constant b/4 is the best possible.

On maximum principles for solutions to nonlinear elliptic ODEs and radial solutions to nonlinear elliptic PDE's via Opial‐type inequalities

“…The following statements contribute to the nonexistence and triviality for solutions to (21) and (22). It can be treated as the problem overdetermined by the condition w(0) = 0.…”

“…the set of conditions:  nd ∶= {1, 2(a), 3(a l ), 4(a l ), 5(a l )} is satisfied, and w is the solution to (21); ( n ) the set of conditions  d ∶= {1, 2(a), 3(b l ), 4(b l ), 5(b l )} is satisfied, and w is the solution to (22).…”

“…The variant of Opial-type inequality due to Beesack and Das. We will use the following variant of Opial-type inequality due to Beesack and Das 5 (see also, e.g., previous studies [22][23][24][25] for some later contributions).…”

Opial-type inequalities for convex functions and associated results in fractional calculus

“…This inequality has been studied by several mathematicians, and several related inequalities have been investigated. For example its generalizations were published by Beesack in 1962 and1971 [3,5], Hua in 1965 [17], Redheffer in 1966 [30], Calvert in 1967 [8], Godunova and Levin in 1967 [13], Maroni in 1967 [22], Boyd and Wong in 1967 [7], Beesack and Das in 1968 [4], Boyd in 1969 [6], Rozanova in 1972 [31], Vrǎnceanu in 1973 [35], Shum in 1974 and1975 [32,33], Hou in 1979 [16], G. Milovanovic and I. Milovanovic in 1980 [23], Lee in 1980 [20], He and Wang in 1981 [14], Yang in 1966 and1983 [37,38], Hong, Yang, and Du in 1982 [15], Lin and Yang in 1985 [21], Qi in 1985 [29], Imoru in 1985 and1986 [9,10], Pachpatte in 1986 and1993 [26,27], Mitrinović and Pečarić in 1988 [24], Hwang and Yang in 1990 [18], and Sinnamon in 1991…”

On some generalizations of dynamic Opial-type inequalities on time scales

“…Cheng-Shyong Lee [18] generalized (1.11) and proved that: If u is an absolutely continuous function on [a, X] with u(a) = 0, then…”