On the mean values of an analytic function

Transcription

1 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.

2 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.

3 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,

4 152 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..

5 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.

6 154 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.

7 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 [2] G. H. Hardy, The mean value of the modulus of an analytic function, Proc. London Math. Soc. 14 (2) (1915), [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), [4] L. R. Sons, Regularity of growth and gaps, J. Math. Anal. Appl. 24 (1968), DEPARTMENT OF MATHEMATICS UNIVERSITY OF ROORKEE ROORKEE , INDIA Reçu par la Rédaction le