ANNALES POLONICI MATHEMATICI LVII.2 (1992) On the mean values of an analytic function by G. S. Srivastava and Sunita Rani (Roorkee) Abstract. Let f(z), z = re iθ, be analytic in the finite disc z < R. The growth properties of f(z) are studied using the mean values I δ (r) and the iterated mean values N δ,k (r) of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z). 1. Let f(z) = n= a nz n, z = re iθ, be analytic in the disc z < R, < R <. For r < R, we set M(r) = max z =r f(z). Then the order ϱ and lower order λ of f(z) are defined as (see [4]) (1.1) log + log + M(r) { ϱ, = λ, λ ϱ, where x = Rr/(R r) and log + t = max{, log t}. When < ϱ <, we define the type T and lower type τ ( τ T ) of f(z) as log + { M(r) T, (1.2) x ϱ = τ. Let m(r) = max n { a n r n } be the maximum term in the Taylor series expansion of f(z) for z = r. If f(z) is of finite order ϱ, then ([1], [3]) (1.3) log m(r) log M(r) as r R. Hence m(r) can be used in place of M(r) in (1.1) and (1.2) for defining ϱ, λ etc. The following mean value of an analytic function f(z) was introduced by Hardy [2]: (1.4) I δ (r) = [J δ (r)] 1/δ = [ 1 ] 1/δ f(re iθ ) δ dθ 1991 Mathematics Subject Classification: 3B1. Key words and phrases: analytic function, maximum term, order, type, mean values.
15 G. S. Srivastava and S. Rani where < δ <. We introduce the following weighted mean of f(z): r ( ) k+1 Ry (1.5) N δ,k (r) = x k dy I δ (y) R y y 2, where x = Rr/(R r) and < k <. In this paper we have studied the growth properties of the analytic function f(z) through its mean values I δ (r) and N δ,k (r). In the sequel, we also derive some convexity properties of these means and also study their relative growths. We shall assume throughout that ϱ <. 2. We now prove Lemma. For every r, < r < R, [x k I δ (r)/(r r)] is an increasing convex function of [x k N δ,k (r)]. P r o o f. From (1.5) we have d[x k I δ (r)/(r r)] d[x k N δ,k (r)] = ri δ (r) RI δ (r) + r R(R r) + k R r, where I δ (r) denotes the derivative of I δ(r) with respect to r. Since R and k are fixed, the last two terms on the right hand side of the above equation are increasing functions of r. Further, it is well known that log I δ (r) is an increasing convex function of log r. Hence the right hand side of the above equation is an increasing function of r and the Lemma follows. Theorem 1. For ϕ(r) = I δ (r), J δ (r) and N δ,k (r), we have { log log ϕ(r) ϱ, (2.1) = λ ϱ <. λ, P r o o f. It is known that for n, a n = 1 i C f(z) dz, zn+1 where C is the circle z = r, < r < R. Hence a n r n 1 f(re iθ ) dθ. Since the right hand side is independent of n, we can choose n suitably to obtain m(r) 1 f(re iθ ) dθ.
Mean values of an analytic function 151 For δ 1, we apply Hölder s inequality to the right hand side. Then m(r) 1 { } 1/δ { } (δ 1)/δ f(re iθ ) δ dθ dθ [ 1 1/δ = f(re iθ ) dθ] δ. Hence m(r) I δ (r). From (1.4) we obviously have I δ (r) M(r). Hence for r > and δ 1, we have (2.2) m(r) I δ (r) M(r). If < δ < 1, then Thus [I 1+δ (r)] 1+δ = f(re iθ ) 1+δ dθ M(r) = M(r)[I δ (r)] δ [M(r)] 1+δ. (2.3) I 1+δ (r) [M(r)] 1/(1+δ) [I δ (r)] δ/(1+δ) M(r). From (2.2) we have, in view of (1.3), log I δ (r) log M(r) as r R, δ 1. f(re iθ ) δ dθ Hence log I (1+δ) (r) log M(r) as r R, < δ < 1. Thus from (2.3) we have log I δ (r) log M(r) as r R, < δ < 1. Combining these two asymptotic relations, we get (2.4) log I δ (r) log M(r) as r R, δ >. From (1.4) and (2.4) we immediately have { log log I δ (r) sup = To prove (2.1) for ϕ(r) = N δ,k (r), we take [ r = R 1 1 ( 1 r )] α R log log J δ (r) { ϱ, = λ. where α > 1 is an arbitrary constant. Then from (1.5) we have r ( ) k+1 Ry N δ,k (r ) = (x ) k dy I δ (y) R y y 2 > (x ) k r r ( ) k+1 Ry dy I δ (y) R y y 2,
152 G. S. Srivastava and S. Rani where x = Rr /(R r ). Since I δ (r) is an increasing function of r, we have (2.5) N δ,k (r ) > I δ(r) k (x ) k x k (x ) k = O(1)I δ (r). It can be easily verified that x /x α and ( )/ 1 as r R. Hence we have { log log N δ,k (r) sup log log I δ (r) (2.6). For the reverse inequality we have from (1.5), (2.7) N δ,k (r) I δ (r)/k. Hence (2.8) log log N δ,k (r) log log I δ (r) Combining (2.6) and (2.8) we get the relation (2.1) for ϕ(r) = N δ,k (r). This proves (2.1) completely. Theorem 2. For < ϱ <, we have (2.9) (2.1) log I δ (r) x ϱ = log N δ,k (r) x ϱ = { T, τ, { T, P r o o f. The relation (2.9) follows easily from (2.4) and the definitions of T and τ. To prove (2.1) we have from (2.7), { log N δ,k (r) sup log I δ (r) (2.11) x ϱ x ϱ. Also, from (2.5) we have Since x /x α as r R, we have τ. log N δ,k (r ) > O(1) + log I δ (r). log N δ,k (r ) (x ) ϱ α ϱ Since α > 1 was arbitrary, we thus have { log N δ,k (r) sup (2.12) x ϱ log I δ (r) x ϱ. log I δ (r) x ϱ. Now combining (2.11) and (2.12), we get (2.1) in view of (2.9). This proves Theorem 2..
Mean values of an analytic function 153 In the next two theorems, we obtain the relative growth of I δ (r) and N δ,k (r). We prove Theorem 3. For the mean values I δ (r) and N δ,k (r) as defined before, we have } { { ϱ sup log[i δ (r)/(r r)n δ,k (r)] ϱ + 1, (2.13) λ λ + 1. P r o o f. From (1.5) we have d dr [xr N δ,k (r)] = x k+1 I δ (r)/r 2 where x = Rr/(R r). Expanding and rearranging the terms on the left hand side, we get N δ,k (r) N δ,k (r) = RI δ (r) r(r r)n δ,k (r) kr r(r r). Integrating on both sides of this equation with respect to r, we get (2.14) log N δ,k (r) = O(1) + R r r I δ (y) dy k log[r/(r r)] y(r y)n δ,k (y) where < r r < R. Since ϱ <, we have from Theorem 1, (2.15) log(r r) log N δ,k (r) =. Now from the Lemma, [I δ (y)/(r y)n δ,k (y)] is an increasing function of y. Hence from (2.14) we have log N δ,k (r) < O(1) + RI δ(r) log(r/r ) (R r)n δ,k (r) or, in view of (2.15), Hence k log[r/(r r)], log N δ,k (r){1 + o(1)} < RI δ(r) log(r/r ) (R r)n δ,k (r). log log N δ,k (r) log[i δ (r)/(r r)n δ,k (r)] In view of (2.1), we get the left hand inequalities of (2.13). To obtain the right hand inequalities of (2.13), we again take arbitrary α > 1 and.
154 G. S. Srivastava and S. Rani r = R[1 (1/α)(1 r/r)]. Then from (2.14), since r > r, log N δ,k (r ) O(1) + R Using (2.15) we have r r I δ (y) dy y(r y)n δ,k (y) k log[r /(R r )] O(1) + RI δ(r) log(r /r) (R r)n δ,k (r) k log[r /(R r )]. (2.16) [1 + o(1)] log N δ,k (r ) RI δ(r) log(r /r) (R r)n δ,k (r) + O(1), or log log N δ,k (r ) log[i δ(r)/(r r)n δ,k (r)] + log log(r /r) + o(1). As before, ()/ 1 and [log log(r /r)]/ 1 as r R. Hence we obtain, on proceeding to its, { log[i δ (r)/(r r)n δ,k (r)] ϱ + 1, λ + 1. This proves Theorem 3. Theorem 4. For < ϱ <, we have I δ (r)/n δ,k (r) (2.17) x ϱ where A = (ϱ + 1) ϱ+1 /ϱ ϱ. P r o o f. From (2.16) we have { AT, Aτ, [1 + o(1)] log N δ,k(r ) (x ) ϱ R log(r /r)i δ (r) (R r)n δ,k (r)(x ) ϱ + o(1). Since log(r /r) = α 1 x and R r αr x = α, where as before x = Rr /(R r ), we get on proceeding to its log N δ,k (r ( ) { ) α 1 sup (x ) ϱ α ϱ I δ (r)/n δ,k (r) α x ϱ. Since α > 1 was arbitrary, we can take α = (ϱ + 1)/ϱ. Hence, using (2.1) we obtain { I δ (r)/n δ,k (r) AT, x ϱ Aτ, where A = (ϱ + 1) ϱ+1 /ϱ ϱ. Thus Theorem 4 follows.
Mean values of an analytic function 155 The authors are thankful to the referee for his valuable comments and suggestions. References [1] R. P. Boas, Entire Functions, Academic Press, New York 1954. [2] G. H. Hardy, The mean value of the modulus of an analytic function, Proc. London Math. Soc. 14 (2) (1915), 269 277. [3] G. P. Kapoor, A note on the proximate order of functions analytic in the unit disc, Rev. Fac. Sci. Univ. d Istanbul Sér. A 36 (1971), 35 4. [4] L. R. Sons, Regularity of growth and gaps, J. Math. Anal. Appl. 24 (1968), 296 36. DEPARTMENT OF MATHEMATICS UNIVERSITY OF ROORKEE ROORKEE 247667, INDIA Reçu par la Rédaction le 3.4.1991