On some inequalities for double and path integrals on general domains\footnote{Dedicated to Sienna and Audrey

Abstract

In this paper, by the use of the celebrated Green’s identity for double and path integrals, we establish some integral inequalities for functions of two variables de…ned on closed and bounded subsets of the plane R2: Some examples for rectangles and disks are also provided. 1. Introduction In paper [1], the authors obtained among others the following results concerning the di¤erence between the double integral on the disk and the values in the center or the path integral on the circle: Theorem 1. If f : D (C;R) ! R has continuous partial derivatives on D (C;R) ; the disk centered in the point C = (a; b) with the radius R > 0; and @f @x D(C;R);1 : = sup (x;y)2D(C;R) @f (x; y) @x <1; @f @y D(C;R);1 : = sup (x;y)2D(C;R) @f (x; y) @y <1; then (1.1) f (C) 1 R2 ZZ D(C;R) f (x; y) dx dy 4 3 R " @f @x D(C;R);1 + @f @y D(C;R);1 # : The constant 4 3 is sharp. We also have (1.2) 1 R2 ZZ D(C;R) f (x; y) dx dy 1 2 R Z (C;R) f ( ) dl ( ) 2R 3 " @f @x D(C;R);1 + @f @y D(C;R);1 # ; 1991 Mathematics Subject Classi…cation. 26D15. Key words and phrases. Double integral, Path integral, Green’s identity, Integral inequalities. 1 2 S. S. DRAGOMIR where (C;R) is the circle centered in C = (a; b) with the radius R > 0 and (1.3) f (C) 1 2 R Z (C;R) f ( ) dl ( ) 2R " @f @x D(C;R);1 + @f @y D(C;R);1 # : In the same paper [1] the authors also established the following Ostrowski type inequality: Theorem 2. If f has bounded partial derivatives on D(0; 1), then (1.4) f (u; v) 1 ZZ D(0;1) f (x; y) dx dy 2 " @f @x D(0;1);1 u arcsinu+ 1 3 p 1 u2 (2 + u) + @f @y D(0;1);1 v arcsin v + 1 3 p 1 v2 (2 + v) # for any (u; v) 2 D (0; 1). For other integral inequalities for double integrals see [2]-[14]. In this paper, by the use of the celebrated Green’s identity for double and path integrals, we establish some integral inequalities for functions of two variables de…ned on closed and bounded subsets of the plane R: Some examples for rectangles and disks are also provided. 2. Main Results Let @D be a simple, closed counterclockwise curve in the xy-plane, bounding a region D. Let L and M be scalar functions de…ned at least on an open set containing D. Assume L and M have continuous …rst partial derivatives. Then the following equality is well known as the Green theorem (see for instance https://en.wikipedia.org/wiki/Green%27s_theorem) (G) Z Z D @M (x; y) @x @L (x; y) @y dxdy = I @D (L (x; y) dx+M (x; y) dy) : Moreover, if the curve @D is described by the function r (t) = (x (t) ; y (t)) ; t 2 [a; b] ; with x, y di¤erentiable on (a; b) then we can calculate the path integral as I @D (L (x; y) dx+M (x; y) dy) = Z b a [L (x (t) ; y (t))x0 (t) +M (x (t) ; y (t)) y0 (t)] dt: By applying this equality for real and imaginary parts, we can also state it for complex valued functions P and Q: INEQUALITIES FOR DOUBLE AND PATH INTEGRALS 3 For a function f : D ! C having partial derivatives on the domain D we de…ne @f;D : D ! C as @f;D (x; y) := (x y) @f (x; y) @x @f (x; y) @y : We need the following identity, [5]: Lemma 1. Let @D be a simple, closed counterclockwise curve in the xy-plane, bounding a region D. Assume that the function f : D ! C has continuous partial derivatives on the domain D: Then 1 2 I @D [(x y) f (x; y) dx+ (x y) f (x; y) dy] Z Z D f (x; y) dxdy (2.1)