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Some generalized Opial-type inequalities. (English) Zbl 0761.26012

In the present note, two Opial-type integral inequalities involving many functions in several independent variables are proved.

Let Ω:=[a 1 ,b 1 ]×[a 2 ,b 2 ]××[a n ,b n ] n be a rectangular region. Denote by x=(x 1 ,x 2 ,,x n ) a general point in Ω and write dx=dx 1 dx n . The main result embodied in Theorem 1 can be re-stated as follows.

Theorem 1. Let f α (α=1,2,,m) and their partial derivatives f 1 α ,f 12 α ,,f 1(n-1) α , and f ˙ α :=f 1n α are all defined and continuous on Ω. Let further F α :[0,)[0,) be any nonnegative differentiable functions, α=1,,m, with F α ' nonnegative, continuous and nondecreasing on [0,). Suppose that

f α (a 1 ,x 2 ,,x n )=f 1 α (x 1 ,a 2 ,x 3 ,,x n )==f 1(n-1) α (x 1 ,,x n-1 ,a n )=0

for all xΩ, α=1,2,,m. Then we have

Ω β=1 m αβ F α (|f α (x)|)F β ' (|f β (x)|)|f ˙ β (x)|dx α=1 m F α Ω |f ˙ α (x)| d x·

Another inequality given in Theorem 2 is obtained under the additional condition

f α (b 1 ,x 2 ,,x n )=f 1 α (x 1 ,b 2 ,x 3 ,,x n )==f 1(n-1) α (x 1 ,,x n-1 ,b n )=0·

Some known inequalities due to G. S. Yang [Tamkang J. Math. 13, 255-259 (1982; Zbl 0516.26009)] established for two-variable functions are contained in the results obtained.

MSC:
26D10Inequalities involving derivatives, differential and integral operators
26D15Inequalities for sums, series and integrals of real functions