On the Mean Modulus of an Analytic Function

For fixed ƒ and r this mean modulus Mt(r;f) as a function of / is continuous, nonnegative, nondecreasing, and is bounded above by the maximum modulus of ƒ on C(r) [l, 2 ] . 1 Therefore the limit of Mt(r;f) exists as /—>0 and /—» oo. This limit is defined to be the mean modulus of ƒ on C(r) of order 0 and of order oo respectively. I t may be shown that the mean modulus of order 0 is the geometric mean of the modulus o f / o n C(r), which is simply evaluated by Jensen's formula, and that the mean modulus of order oo is the maximum modulus of ƒ on C(r) [l, 2] . Thus Mt(r;f) is defined for all parameters / in the compact infinite interval 0 = ^ ^ oo. For fixed ƒ and t ( 0 ^ / ^ oo) the mean modulus Mt(r\f) as a function of r in the interval 0 ^ r < l is continuous, nonnegative, nondecreasing, and, except for the limiting parameters 0 and oo, possesses a continuous derivative with respect to r[l, 3] . Moreover, its logarithm is a nondecreasing convex function of log r (for t = oo this is the Hadamard three-circle theorem) [l, 3] . We shall be concerned here with the convexity of the mean modulus Mt{r\f) as a function of r. Let T(f) be the set of all parameters / in the compact infinite interval 0 ^t^ oo such that Mt(r;f) is a convex function of r in the interval O ^ X l . Since Mt(r\f) is continuous with respect to the parameter / and since any function which is the limit of convex functions is also convex, the set T(J) is a closed and hence compact subset of the parameter interval 0 ^t^ oo. The set T(J) need not, however, coincide with the entire parameter interval and indeed