On some new integral inequalities in two independent variables
The author considers some new two independent variables analogs of the known Wirtinger and Opial type one-variable integral inequalities due to himself [Tamkang J. Math. 17, No.1, 69-73 (1986; Zbl 0623.26013)] and \textit{J. Traple} [Zesz. Nauk. Uniw. Jagiellon., 252, Pr. Mat. 15, 159-168 (1971; Zbl 0285.34014)]. Let \(p(x,y)\geq 0\), \(f_ r(x,y)\), \(D_ 1f_ r\) and \(D_ 1D_ 2f_ r\) (where \(D_ 1=\partial /\partial x\), \(D_ 2=\partial /\partial y)\) be real-valued continuous functions defined on \(\Delta\), with \(f_ r(a,y)=f_ r(b,y)=D_ 1f_ r(x,c)=D_ 1f_ r(x,d)\equiv 0\) for (x,y)\(\in \Delta\), here \(\Delta:=[a,b]\times [c,d].\) Let further \(m_ r\geq 1\) be real numbers, \(r=1,...,n\). The main results of the paper can be simply restated as: \[ \int^{b}_{a}(\int^{d}_{c}p(F)^ 2dy)dx \leq \frac{1}{n}KP_ 1\int^{b}_{a}(\int^{d}_{c}G_ 2dy)dx\text{ (Theorem 1);} \] \[ \int^{b}_{a}(\int^{d}_{c}pFG_ 1dy)dx \leq (KP_ 2)^{1/2}\int^{b}_{a}(\int^{d}_{c}G_ 2dy)dx\text{ (Theorem 2),} \] where K is a constant depending on \(a,b,c,d,n,m_ 1,...,m_ n\), and \[ F(x,y):=(\prod^{n}_{r=1}| f_ r(x,y)|^{m_ r})^{1/n};\quad P_ k:=\int^{b}_{a}(\int^{d}_{c}p^ k(x,y)dy)dx, \] \[ G_ k(x,y):=\sum^{n}_{r=1}| D_ 1D_ 2f_ r(x,y)|^{km_ r}\quad (k=1,2). \] \{Reviewer's note: The indices of the Hölder's inequality used on page 379 to derive (18) from (17) should be \(m_ r\), \(m_ r/(m_ r-1)\), for \(r=1,2,...,n\). Hence the proof for Theorem 1 given here is not suitable for the case when \(m_ r=1\) holds for some r, \(r=1,...,n.\)\}
scientific article; zbMATH DE number 4040207
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