周 偉

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

1 預(yù)備知識(shí)

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(hào).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(dāng)

(5)

時(shí),結(jié)論是精確的.

從而有

(6)

考慮z取實(shí)值.取z=r(0≤r<1).則當(dāng)r=0時(shí),(6)式的分母是正的,從而對(duì)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(jì).

當(dāng)f(z)為(5)式時(shí),等號(hào)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

參考文獻(xiàn):

[1] SRIVASTAVA H M, PATEL J. Some subclasses of multivalent functions involving a certain linear operator[J].J Math Anal Appl, 2005,310(1):209-228.

[2] LIU J L, SRIVASTAVA H M. Subclasses of meromorphically multivalent functions associated with a certain linear operator[J].Math Comput Modelling, 2004,39(1):35-44.

[3] LIU J L. SRIVASTAVA H M. Classes of meromorphically multivalent functions associated with the generalized hypergeometeic function[J].Math Comput Modelling, 2004,39(1):21-34.

[4] SRIVASTAVA H M, HOSSEN H M, AOUF M K. A unified presentation of some classes of meromorphically multivalent functions[J]. Comput Math Applic, 1999,38(11-12):63-70.

[5] LIU J L, SRIVASTAVA H M. A linear operator and associated families of meromorphically multivalent functions [J]. J Math Anal Appl, 2001,259(2):566-581.

[6] YANG D G. On meromorphic starlike multivalent functions[J].Chin Quart J Math, 1993,8(3):88-93.

[7] 李書海.關(guān)于β級(jí)預(yù)星像函數(shù)的一個(gè)子類[J].陜西師范大學(xué)學(xué)報(bào)：自然科學(xué)版, 1999,51(S1):49-52.

[8] SRIVASTAVA H M, MISHRA A K. Applications of fractional calculus to parabolic starlike and uniformly convex functions[J]. Comput Math Appl, 2000,39(3-4):57-69.

[9] DZIOK J, SRIVASTAVA H M. Classes of analytic functions associated with the generalized hypergeometric function[J]. Appl Math Comput, 1999,103(1):1-13.

[10] DZIOK J, SRIVASTAVA H M. Certain subclasses of analytic associated with the generalized hypergeometric function[J].Integral Transfom Spec Funct, 2003,14(1):7-18.