亞純多葉函數(shù)的一子類性質(zhì)

周 偉

(淮陰師范學(xué)院數(shù)學(xué)科學(xué)學(xué)院,中國 淮安 223300)

1 預(yù)備知識

定義1設(shè)∑P表示形如

且在E=z:0

設(shè)函數(shù)φp(a,c;z)為

(x)0=1,(x)k=x(x+1)(x+2)…(x+k-1) ,k∈N,

這里(x)k為Pochhammer記號.它可以表示成超幾何函數(shù)[9-10]

定義2設(shè)f∈∑p,定義關(guān)于∑p的線性算子Lp(a,c)如下

Lp(a,c)f(z)=φp(a,c;z)f(z).

可證明線性算子Lp(a,c)滿足

(1)

z(Lp(a,c)f(z))′=aLp(a+1,c)f(z)-(a+p)Lp(a,c)f(z).

定義3若f∈∑p且滿足

其中z∈E,-1≤B

(2)

2 主要結(jié)論

假定a>0,c>0且0≤B<1.

(3)

其中

(4)

當

(5)

時,結(jié)論是精確的.

從而有

(6)

考慮z取實值.取z=r(0≤r<1).則當r=0時,(6)式的分母是正的,從而對于所有的r(0

從而(3)式成立.

反之,由(3)式可得:

由定理1可得下面精確的系數(shù)估計.

當f(z)為(5)式時,等號成立.

(7)

其中Ck=kΓk+p(a)[k(1+B)+p(A-B)],k=p,p+1,…;p∈N，且Γm(a)由(4)式給出.

(8)

由(5)式給出的f(z)為(7),(8)式的極值函數(shù).

從而可得(7),(8)式成立.定理得證.

(1)f在圓盤z

(9)

其中

(10)

且Γm(a)由(4)式給出.

(2)f在圓盤z

(11)

其中

(12)

且Γm(a)由(4)式給出.由(5)式給出的f(z)為這些結(jié)論的極值函數(shù).

證(1)由定義(2)式可得

(13)

則

從而可得(9)式成立.

由(13)式可得:z

(2) 由(2)式得

(14)

則

由(14)式可得:z

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