Mean Value Derivatives

Let L(f;x,y;r), A(f;x,y;r) be the mean values of a function f(x,y) of two real variables on the perimeter and on the interior, respectively, of a circle of center (x,y) and radius r. The limits limr→0 L(f;x,y;r) - f(x,y)/r2 = f'(x,y) limr→0 A(f;x,y;r) - f(x,y)/r2 = f'D(x,y) are called Mean Value Derivatives of f(x,y). This paper is concerned with the investigation of functions with mean value derivatives. These derivatives are essentially generalizations of LaPlace's operator, and, as such, were investigated by Blaschke and Privaloff. In addition Zaremba has investigated another form of generalized LaPlacian, and Plancherel has investigated a generalization of Beltrami's parameter. Many of the results obtained for these last two operators hold true for mean value derivatives. Chapter I contains some results relating to the mean value derivative as given by eqn. (1) while Chapter II is a similar treatment of eqn.(2.). Most of the results given in these two chapters are known for at least one of the four operators, i.e. those of Blaschke, Privaloff, Zaremba, and Plancherel. Chapter III discusses briefly uniform mean value derivatives. Chapter IV is devoted to the use of potential theory in the subject and Chapter V to higher derivatives. Chapter VI is concerned with further problems on the subject and Chapter VII contains a summary of the results of the authors mentioned above. The principle new results obtained are as follows: (1) If f'0(x,y) exists then so does f'(x,y). This is a generalization of a result due to Kozakiewicz, who assumed continuity of f. Tb.is assumption is not necessary. (2) If (i)f(x,y)is continuous, (ii) f'(x,y) exists and is bounded, (iii) f'(x,y)=o almost everywhere on a domain D, then f(x,y) is harmonic on D. (3) If f(x,y) is a logarithmic potential function for which the density of the mass distribution exists at a point P then f'(P) exists. (4) Expansions in powers of r2 are obtained for the means L(f;x,y;r), A(f;x,y;r) in which the coefficients involve the higher mean value derivatives of in a manner analogous to Taylor's Theorem.