GUO Yi-bing,WANG Ping-an

(College of Sciences,Xijing University,Shaanxi,710123,P.R.China)

Abstract:In this paper,we mainly discuss the problem of estimating the n-th derivative for bounded regular vanishing functions.The estimation of the n-th derivative for the function is deduced by the 1-th and 2-th derivative.

Key words:Vanishing function;bounded regular functions;estimation;derivative

§1. Introduction

Consider the following family of functions:

Goluzin obtained the derivative estimations for the family B of functions.

Let ?(z)=c0+c1z+ ···+cnzn+ ···be regular in|z|<1 and|?(z)|<1.Then

Dieudonne also obtained this result for used the different method.In 1973,Shaffer generalized[6]the above conclusions to bounded regular vanishing functions

and got the following conclusion:

Let ?(z)∈ Bn?1(n ≥ 1).Then we have

In 1983,Pan and Liao[1]have obtained the 2-th derivative estimation for the above functions.

Let B={?(z)|?(z)=c+c1z+ ···+cnzn+ ···,and|?(z)|<1}.Then we have

where ρnis the smallest positive root in(0,1)of

In this paper,we mainly discuss the problem of estimating the 3-th,4-th and the n-th derivatives for bounded regular vanishing functions.

§2.Main Result

Lemma 2.1[4]Let ?(z)=c0+c1z+···+cnzn+···be regular in|z|<1 and|?(z)|<1.Then we have

Theorem 2.1 Let ?(z)∈ Bn+1.Then we have

ρnis the smallest positive root in(0,1)of

Let g(z)be regular in|z|<1,and|g(z)|<1.So g(z)satisfies(1.1),(1.2).Now substituting these into(2.2),let x=|g(z)|,|z|=r.Then we have

So we complete the proof.Equation holds if and only if ?(z)=ei?zn+2.So the estimate is accurate.

By theorem 2.1,we can get the following theorem.

Theorem 2.2 Let ?(z)∈ B2n?2.Then

Note that P2(r)is continuous on(0,1).By the properties of continuous function and the local security number of function limit,we can get that P2(r)=0 has the smallest positive root in(0,1).

Then we proof this theorem.

Now substituting(2.2)into the above formula,then we have

By the above formula we can get the minimum value whenequivalents to P2(r)=0(P2(r)is obtained by H0(x)=0 whenis the smallest positive root in(0,1).

So we complete the proof.

Thus we solve the problem of estimating the n-th derivatives for bounded regular vanishing functions.