Research Article Properties of Certain Subclass of Multivalent Functions with Negative Coefficients

Transcription

1 International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID , 21 pages doi: /2012/ Research Article Properties of Certain Subclass of Multivalent Functions with Negative Coefficients Jingyu Yang 1, 2 and Shuhai Li 1 1 Department of Mathematics, Chifeng University, Inner Mongolia, Chifeng , China 2 School of Mathematical Sciences, Dalian University of Technology, Liaoning , China Correspondence should be addressed to Shuhai Li, lishms66@sina.com Received 26 December 2011; Revised 26 March 2012; Accepted 27 March 2012 Academic Editor: Marianna Shubov Copyright q 2012 J. Yang and S. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Making use of a linear operator, which is defined by the Hadamard product, we introduce and study a subclass Y a,ca, B; p, λ, α of the class A p. In this paper, we obtain the coefficient inequality, distortion theorem, radius of convexity and starlikeness, neighborhood property, modified convolution properties of this class. Furthermore, an application of fractional calculus operator is given. The results are presented here would provide extensions of some earlier works. Several new results are also obtained. 1. Introduction Let A p denote the class of functions fz z p ( ) akpz kp p N {1, 2,...} 1.1 which are analytic and p-valent in the open unit disc U {z : z < 1} on the complex plane C. We denote by S pa, B and K p A, B 1 B<A 1 the subclass of p-valent starlike functions and the subclass of p-valent convex functions, respectively, that is, see for details 1, 2 S pa, B { fz A ( p ) : zf z 1 Az pfz 1 Bz } z U; 1 B<A 1,

2 2 International Journal of Mathematics and Mathematical Sciences K p A, B { fz A ( p ) : 1 p zf z 1 Az pf z 1 Bz } z U; 1 B<A A function f A p is said to be analytic starlike of order ξ if it satisfies { } zf z Re >ξ 1.3 pfz for some ξ 0 ξ<1 and for all z U. Further, a function f A p is said to be analytic convex of order ξ if it satisfies { 1 Re p zf } z pf >ξ 1.4 z for some ξ0 ξ<1and for all z U. For fz z p a kpz kp and gz z p b kpz kp, the Hadamard product or convolution is defined by ( ) f g z z p a kp b kp z kp ( g f ) z. 1.5 The linear operator L p a, c is defined as follows see Saitoh 3: L p a, cfz ϕ p a, c; z fz ( fz A ( p )), 1.6 and ϕ p a, c; z is defined by ϕ p a, c; z z p a k z kp c k ( a R; c \ z 0 ; z 0 : {0, 1, 2,...}), 1.7 where v k is the pochharmmer symbol defined in terms of the Gamma function by ν n Γν n Γν { 1, n 0, νν 1 ν n 1, n N. 1.8 The operator L p a, c was studied recently by Srivastava and Patel 4. It is easily verified from 1.6 and 1.7 that z ( L p a, cfz ) alp a 1,cfz ( a p ) L p a, cfz. 1.9

3 International Journal of Mathematics and Mathematical Sciences 3 Moreover, for fz Ap, ( ) zf z L p a, afz fz, L p p 1,p fz, p 1.10 ( ) L p n p, 1 fz D np 1 ( ) fz n> p, where D np 1 fz denotes the Ruscheweyh derivative of a function fz Az of order n p 1 see 5. Aouf, Silverman and Srivastava 6 introduced the class of P a,ca, B; p, λ by use of linear operator L p a, c, further investigated the properties of this class. In 7, SokóŁ investigated several properties of the linear Aouf-Silverman-Srivastava operator and furthermore obtained the corresponding characterizations of multivalent analytic functions which were studied by Aouf et al. in paper 6. The properties of multivalent functions with negative coefficients were studied in References 11, 12 gave the results of the univalent function with negative coefficients. In this paper, we will use operator L p a, c to define a new subclass of A p as follows. For p N, a > 0, c > 0, α 0 and for the parameters λ, A, and B such that 1 B<A 1, 1 B<0, 0 λ<p, 1.11 we say that a function fz A p is in the class Y a,ca, B; p, λ, α if it satisfies the following subordination: (( 1 Lp a, cfz ) p λ z p 1 ( Lp a, cfz ) ) α 1 pz p 1 λ 1 Az 1 Bz z U, 1.12 or equivalently, if the following inequality holds true: ( Lp a, cfz ) /z p 1 α ( L p a, cfz ) /pz p 1 1 p (Lp B( a, cfz ) /z p 1 α ( L p a, cfz ) /pz p 1 ) 1 ( pb A B ( p λ )) < From the above definition, we can imply that the function class P a,ca, B; p, λ in 6 is the special case of Y a,ca, B; p, λ, α in our present paper because Y a,ca, B; p, λ, 0 P a,ca, B; p, λ. Since Y a,ca, B; p, λ, 0 P a,ca, B; p, λ, then, like 6, we have the following subclasses which were studied in many earlier works: 1 Y np,1 1, 1; p, λ, 0 Q np 1λ 0 λ<1; n> p; p N Aouf and Darwish 8, 2 Ya,aA, B; p, 0, 0 P p, A, B Shukla and Dashrath 9, 3 Ya,a1, 1; p, λ, 0 F p 1,λ0 λ<p; p N Lee et al. 10, 4 Ya,a β, β; 1,λ,0 P λ, β 0 λ<1; 0 <β 1 Gupta and Jain 11,

4 4 International Journal of Mathematics and Mathematical Sciences 5 Y np,1 1, 1; 1,λ,0 Q nλ n N 0 Sarangi 12. : N {0}; 0 λ < 1 Uralegaddi and The purpose of this paper is to give various properties of class Y a,ca, B; p, λ, α. We extend the results of basic paper Necessary and Sufficient Condition for fz Y a,ca, B; p, λ, α Theorem 2.1. Let the function fz A p be given by 1.1, thenfz Y a,ca, B; p, λ, α if and only if ( ) φ k p, α; B, a, c akp A B, 2.1 where φ k ( p, α; B, a, c ) ( p α ) 1 B ( k p ) ak p ( p λ ) c k. 2.2 Proof. For the sufficient condition, let fz z p a kp z kp, we have ( Lp a, cfz ) /z p 1 α ( L p a, cfz ) /pz p 1 1 p (Lp B( a, cfz ) /z p 1 α ( L p a, cfz ) /pz p 1 ) ( pb A B ( p λ )) ( B p ( L p a, cfz ) αe iθ ( L p a, cfz ) pz p 1 p 2 z p 1 p ( L p a, cfz ) αe iθ ( L p a, cfz ) ) pz p 1 p ( pb A B ( p λ )), z p then p ( L p a, cfz ) αe iθ ( Lp a, cfz ) pz p 1 p 2 z p 1 B (p ( L p a, cfz ) αe iθ ( Lp a, cfz ) ) pz p 1 p ( pb A B ( p λ )) z p 1 p ( k p ) akp a k z kp 1 αe iθ c k ( ) k p akp a k z kp 1 c k Bp ( k p ) akp a k z kp 1 αbe iθ c k ( ) k p akp a k z kp 1 c k pa B ( p λ ) z p 1

5 International Journal of Mathematics and Mathematical Sciences 5 p ( k p ) akp a k z kp 1 αe iθ ( ) k p akp a k z kp 1 c k c k B p ( k p ) akp a k z kp 1 ( ) α B k p akp a k z kp 1 c k c k pa B ( p λ ) z p 1 ( p α ) ( ) k p 1 B a kp a k pa B ( p λ ) z U. 2.4 c k By the maximum modulus theorem, for any z U, we have p ( L p a, cfz ) αe iθ ( Lp a, cfz ) pz p 1 p 2 z p 1 B (p ( L p a, cfz ) αe iθ ( Lp a, cfz ) ) pz p 1 p ( pb A B ( p λ )) z p 1 ( p α ) ( ) k p 1 Bakp a k pa B ( p λ ) c k ( ) φ k p, α; B, a, c akp A B 0, 2.5 this implies fz Y a,ca, B; p, λ, α. For necessary condition, let fz Y a,ca, B; p, λ, α be given by 1.1, then ( Lp a, cfz ) z p 1 p ( ) k p akp a k z k, c k 2.6 from 1.13 and 2.6,wefindthat ( Lp a, cfz ) /z p 1 α ( L p a, cfz ) /pz p 1 1 p (Lp B( a, cfz ) /z p 1 α ( L p a, cfz ) ) /pz p 1 1 ( pb A B ( p λ )) ) ( k p akpak /c k z k α (( ) ) k p /p akpak /c k z k A B ( p λ ) B ) ( kp akp ak /c k z k αb (( ) ) kp /p akp ak /c k z k <1. 2.7

6 6 International Journal of Mathematics and Mathematical Sciences Re Now, since Rez z for all z, we have ( ) k p akpak /c k z k α (( ) ) k p /p akpak /c k z k ( ) kp a ak kp /c k z k αb (( ) ) kp /p a ak kp /c k z k A B ( p λ ) B < We choose value of z on the real axis so that the expression L p a, cfz /z p 1 is real. Then, we have Re ) ( kp akpak /c k z k α (( ) ) kp /p akpak /c k z k A B ( p λ ) B ) ( kp akpak /c k z k αb (( ) ) kp /p akpak /c k z k ) ( kp akpak /c k z k α (( ) ) kp /p akpak /c k z k A B ( p λ ) B ( ) kp a ak kp /c k z k αb (( ) ) kp /p a ak kp /c k z k, 2.9 ( ) k p a ak kp /c k z k α (( ) ) k p /p a ak kp /c k z k A B ( p λ ) B ) ( kp akpak /c k z k αb (( ) ) kp /p akpak /c k z k ( ) kp akpak /c k z k α A B ( p λ ) B (( ) ) k p /p akpak /c k z k ( ) kp akpak /c k z k αb (( ) ) kp /p akpak /c k z k < Letting z 1 throughout real values in 2.10,weget ) ) (( k p /p akpak /c k ( ) kp akpak /c k αb ) ) (( kp /p akpak /c k ( ) k p akpak /c k α A B ( p λ ) B < So we have p( k p ) akpak /c k α ) ( k p akpak /c k A B ( p λ ) B p( k p ) akpak /c k αb ( ) k p akpak /c k < 1, 2.12

7 International Journal of Mathematics and Mathematical Sciences 7 it is ( ) ( ) p α 1 B k p a kp a k pa B ( p λ ). c k 2.13 This completes the proof of the theorem. Corollary 2.2. Let the function fz A p be given by 1.1. Iffz Y a,ca, B; p, λ, α, then pa B ( p λ ) c akp k ( ) ( ) ( ) k, p N p α 1 B k p ak The result is sharp for the function given by fz z p pa B ( p λ ) c k ( ) ( ) z kp p α 1 B k p ak 3. Distortion Inequality of Class Y a,ca, B; p, λ, α Theorem 3.1. If a function fz defined by 1.1 is in the class Y a,ca, B; p, λ, α,then ( p! ( p m )! cpa B ( p λ ) ) p! a ( p α ) 1 B ( p 1 m )! z z p m f m z ( p! cpa B ( p λ ) ) p! ( ) p m! a ( pα ) 1 B ( p1 m )! z z p m ( z U; a c>0; m N 0 ; p>m ). 3.1 The result is sharp for the function fz given by fz z p ca B( p λ ) a1 B ( p α ) z1p ( p N ). 3.2 Proof. Since fz z p a kp z kp, then f m z p! ( p m )! z p m ( ) k p! ( ) a kp z kp m. 3.3 k p m! From fz Ya,cA, B; p, λ, α, usingtheorem 2.1, we have a ( p α ) 1 B ( 1 p ) ( )( ) ( ) p α k p 1 Bak k p! akp cpa B ( p λ )( p 1 )! pa B ( p λ ) c k akp 1 3.4

8 8 International Journal of Mathematics and Mathematical Sciences which readily yields ( ) cpa B ( p λ ) p! k p! akp a ( p α ) B By 3.3 and 3.5, we can imply Radius of Starlikeness and Convexity Theorem 4.1. Let the function fz defined by 1.1 be in the class Y a,ca, B; p, λ, α. Then, fz is starlike of η 0 η<1 in z <r η p, k, α, A, B, a, c where { ( ) ( )} 1/k ( ) p 1 η φk p, α; B, a, c r η p, k, α, A, B, a, c inf A B [ k p ( 1 η )] 4.1 and φ k p, α; B, a, c is defined by 2.1. Proof. We must show that zf z pfz 1 < 1 η for z <r ) η( p, k, α, A, B, a, c. 4.2 Since zf z pfz 1 k akpz k p p akpz k k akp z k p p, akp z k 4.3 to prove 4.1, itissufficient to prove k akp z k p p akp z k < 1 η. 4.4 It is equivalent to [ ( )] k p 1 η akp p ( 1 η ) z k By Theorem 2.1, we have ( ) φ k p, α; B, a, c a kp 1, A B 4.6

9 International Journal of Mathematics and Mathematical Sciences 9 hence 4.5 will be true if [ ( )] k p 1 η akp p ( 1 η ) z k φ ( ) k p, α; B, a, c akp. A B 4.7 It is equivalent to { ( ) ( )} 1/k p 1 η φk p, α; B, a, c z A B [ k p ( 1 η )]. 4.8 This completes the proof. Theorem 4.2. Let the function fz defined by 1.1 be in the class Y a,ca, B; p, λ, α. Then, fz is convex of ξ0 ξ<1 in z <r ξ p, k, α, A, B, a, c where r ξ ( p, k, α, A, B, a, c ) inf { p 2 1 ξφ k ( p, α; B, a, c ) ( k p ) A B [ k p1 ξ ]} 1/k 4.9 and φ k p, α; B, a, c is defined by 2.1. Proof. It sufficient to show that zf z pf z p 1 p < 1 ξ for z <r ) ξ( p, k, α, A, B, a, c Since zf z pf z p 1 p k( k p ) a kp z k p 2 p( k p ) akpz k k( k p ) a kp z k p 2 p( k p ) a, kp z k 4.11 to prove 4.9, itissufficient to prove k( k p ) akp z k p 2 p( k p ) akp z k < 1 ξ It is equivalent to ( )[ ] k p k p1 ξ akp z k 1. p 2 1 ξ By Theorem 2.1, we have ( ) φ k p, α; B, a, c a kp 1, A B

10 10 International Journal of Mathematics and Mathematical Sciences hence 4.13 will be true if ( )[ ] k p k p1 ξ akp z k φ ( ) k p, α; B, a, c akp. p 2 1 ξ A B 4.15 It is equivalent to z { p 2 1 ξφ k ( p, α; B, a, c ) ( k p ) A B [ k p1 ξ ]} 1/k This completes the proof. 5. δ-neighborhood of Y a,ca, B; p, λ, α Based on the earlier works by Aouf et al. 6, Altintas et al , and Aouf 16, we introduce the δ-neighborhood of a function fz A p of the form 1.1 and present the relationship between δ-neighborhood and corresponding function class. Definition 5.1. For δ > 0, a > 0, c>0, the δ-neighborhood of a function fz A p is defined as follows: { N ( ) δ f g : gz z p bkpz kp A ( p ), t k bkp akp <δ }, 5.1 where t k ( p α ) 1 B ( k p ) ak pa B ( p λ ) c k. 5.2 Theorem 5.2. Let the function fz defined by 1.1 be in the class Y a1,ca, B; p, λ, α. Then, N ( ) ( ) δ f Y a,c A, B; p, λ, α ( ) a 1 This result is the best possible in the sense that δ cannot be increased. Proof. Let fz Y a1,ca, B; p, λ, α, then by Theorem 2.1, we have ( p α ) 1 B ( k p ) a 1k pa B ( p λ ) c k a kp 1 5.4

11 International Journal of Mathematics and Mathematical Sciences 11 which is equivalent to ( p α ) 1 B ( k p ) ak pa B ( p λ ) c k a kp a a For any gz z p bkpz kp N ( ) δ f, 5.6 we find from 5.1 that ( p α ) 1 B ( k p ) ak pa B ( p λ ) c k bkp akp δ. 5.7 By 5.5 and 5.7, weget ( ) ( ) p α 1 B k p ak bkp pa B ( p λ ) c k ( ) ( ) ( ) ( ) p α 1 B k p ak pa B ( p λ ) p α 1 B k p ak akp c k pa B ( p λ ) bkp akp c k a δ 1. a By Theorem 2.1, it implies that gz Y a,ca, B; p, λ, α. To show the sharpness of the assertion of Theorem 5.2, we consider the functions fz and gz given by fz z p cpa B ( p λ ) a 1 ( p α ) 1 B ( p 1 )zp1 Y ( ) a1,c A, B; p, λ, α, ( gz z p cpa B ( p λ ) ( ) ) a 1 ( p α ) 1 B ( p 1 ) cpa B p λ δ a ( p α ) 1 B ( 1 p ) z p1, 5.9 where δ >δ 1/a 1.

12 12 International Journal of Mathematics and Mathematical Sciences Since ( )( ) p α p 1 1 Ba1 b 1p a 1p pa B ( p λ ) c 1 ( )( ) p α p 1 1 Ba1 pa B ( p λ ) cpa B ( p λ ) δ c 1 a ( p α ) 1 B ( 1 p ) 5.10 δ, so gz N f. δ But ( )( ) p α p 1 1 Ba1 pa B ( p λ ) c 1 a δ a 1 > 1, [ cpa B ( p λ ) ( ) ] a 1 ( p α )( p 1 ) 1 B cpa B p λ δ a ( p α ) 1 B ( 1 p ) 5.11 by Theorem 2.1 gz Y a,ca, B; p, λ, α. 6. Properties Associated with Modified Hadamard Product Following early works by Aouf et al. in 6, we provide the properties of modified Hadamard product of Y a,ca, B; p, λ, α. For the function f j z z p ( ) akp,jz kp j 1, 2; p N, 6.1 the modified Hadamard product of the functions f 1 z and f 2 z was denoted by f 1 f 2 z and defined as follows: ( ) f1 f 2 z z p akp,1 akp,2z kp ( ) f 2 f 1 z. 6.2 Theorem 6.1. Let f j z j 1, 2 given by 6.1 be in the class Y a,ca, B; p, λ, α, then ( f1 f 2 ) z Y a,c ( A, B; p, γ, α ), 6.3 where γ : p cpa B ( p λ ) 2 a ( p α ) 1 B ( 1 p ). 6.4

13 International Journal of Mathematics and Mathematical Sciences 13 The result is sharp for the functions f j z j 1, 2 given by f j z z p cpa B ( p λ ) a ( p α ) 1 B ( 1 p )zp Proof. By Theorem 2.1, we need to find the largest γ such that ( p α ) 1 B ( k p ) ak pa B ( p γ ) c k akp,1 akp, Since f j z Y a,ca, B; p, λ, α, j 1, 2, then we see that ( p α ) 1 B ( k p ) ak pa B ( p λ ) c k akp,j 1 ( j 1, 2 ). 6.7 By Cauchy-Schwartz inequality, we obtain ( p α ) 1 B ( k p ) ak pa B ( p λ ) c k akp,1 akp, This implies that we only need to show that akp,1 akp,2 p γ akp,1 akp,2 p λ k N 6.9 or equivalently that akp,1 akp,2 p γ p λ By making use of the inequality 6.8, itissufficient to prove that pa B ( p λ ) c k ( p α ) 1 B ( k p ) ak p γ p λ k N From 6.11, we have γ p pa B ( p λ ) 2 ck ( p α ) 1 B ( k p ) ak k N. 6.12

14 14 International Journal of Mathematics and Mathematical Sciences Define the function Φk by Φk : p pa B ( p λ ) 2 ck ( p α ) 1 B ( k p ) ak k N We see that Φk is an increasing function of k. Therefore, we conclude that γ Φ1 p pa B ( p λ ) 2 c1 ( p α ) 1 B ( k p ) a By using arguments similar to those in the proof of Theorem 6.1, we can derive the following result. Theorem 6.2. Let f 1 z defined by 6.1 be in the class Y a,ca, B; p, λ, α, f 2 z defined by 6.1 be in the class Y a,ca, B; p, γ, α,then ( f1 f 2 ) z Y a,c ( A, B; p, ξ, α ), 6.15 where ξ : p cpa B( p λ )( p γ ) a ( p α ) 1 B ( 1 p ) The result is sharp for the functions f j z j 1, 2 given by f 1 z z p cpa B ( p λ ) a ( p α ) 1 B ( 1 p )zp1 ( ) p N, f 2 z z p cpa B ( p γ ) a ( p α ) 1 B ( 1 p )zp1 ( ) p N Theorem 6.3. Let f j z j 1, 2 defined by 6.1 be in the class Y a,ca, B; p, λ, α, then the function hz defined by hz z p ( a kp,1 2 a ) kp, belongs to the class Y a,ca, B; p, χ, α where 2pcA B ( p λ ) χ : p a ( p α ) 1 B ( 1 p ) This result is sharp for the functions given by 6.5.

15 International Journal of Mathematics and Mathematical Sciences 15 Proof. By Theorem 2.1, we want to find the largest χ such that ( p α ) 1 B ( k p ) ak pa B ( p χ ) c k ( akp,1 2 akp,2 2 ) Since f j z Y a,ca, B; p, λ, α j 1, 2, we readily see that ( p α ) 1 B ( k p ) ak pa B ( p χ ) c k a kp,j 1 ( j 1, 2 ) From 6.21, we have ( ) 21 p α B 2 ( k p ) 2 ( ) 2 ak p 2 A B 2( p χ ) akp,j c k ( j 1, 2 ), 6.22 then we have ( ) 21 p α B 2 ( k p ) 2 ( ) 2 ak ( p 2 A B 2( p χ ) akp,1 2 ) akp, c k 6.23 From 6.23, if we want to prove 6.20, itissufficient to prove there exists the largest χ such that ( )( ) 1 p α k p 1 p χ Bak 2pA B ( p λ ) k N, ck that is χ p 2pA B( p λ ) 2 ck ( p α )( k p ) 1 Bak k N Now we define Ψk by Ψk p 2pA B( p λ ) 2 ck ( p α )( k p ) 1 Bak k N We see that Ψk is an increasing function of k. Therefore, we conclude that 2pcA Bp λ2 χ Ψ1 p a ( p α )( k p ) 1 B 6.27 which completes the proof of Theorem 6.1.

16 16 International Journal of Mathematics and Mathematical Sciences 7. Application of Fractional Calculus Operator References have studied the fractional calculus operators extensively. In this part, we only investigate the application of fractional calculus operator which was defined by 6 in the class of Y a,ca, B; p, λ, α. Definition 7.1 see 6. The fractional integral of order μ is defined, for a function fz,by D μ z fz 1 z Γ ( μ ) 0 fz z ξ 1 μ dξ ( μ>0 ), 7.1 where the function fz is analytic in a simply connected domain of the complex z-plane containing the origin and the multiplicity of z ξ μ 1 is removed by requiring logz ξ to be real when z ξ>0. Definition 7.2 see 6. The fractional integral of order μ is defined, for a function fz, by D μ zfz 1 z Γ ( 1 μ ) fz 0 z ξ μ dξ ( ) 0 μ<1, 7.2 where the function fz is constrained, the multiplicity of z ξ μ Definition 7.1. is removed as in In our investigation, we will use the operators J δ,p defined by cf, ( Jδ,p f ) z : δ p z p z 0 t δ 1 ftdt ( f A ( p ) ; δ> p; p N ) 7.3 as well as D μ z for which it is well known that see 23 D μ zz ρ Γ ( ρ 1 ) Γ ( ρ μ 1 )zρ μ ( ρ> 1; μ R ) 7.4 in terms of Gamma. Lemma 7.3 see Chen et al. 18. For a function fz Ap, D μ z{( Jδ,p f ) z } Γ ( p 1 ) Γ ( p μ 1 )zp μ ( δ p ) Γ ( p 1 ) ( ) ( ) δ p Γ k p 1 ( ) ( )a kp z kp μ, δ k p Γ k p μ 1 J δ,p (Dz{ μ } ) fz ( ) ( )z p μ δ p μ Γ p μ ( ) ( ) δ p Γ k p 1 ( ) ( )a kp z kp μ δ k p μ Γ k p μ 1 μ R; δ> p; p N provided that no zeros appear in the denominators in 7.5.

17 International Journal of Mathematics and Mathematical Sciences 17 Remark 7.4. Throughout this section, we assume further that a c>0. Theorem 7.5. Let function defined by 1.1 be in the class Y a,ca, B; p, λ, α, then D μ z D μ z {( Jδ,p f ) z } Γ ( p 1 ) ( Γ ( cp ( δ p ) A B ( p λ ) ) p μ 1 ) z pμ 1 a ( p α )( p δ 1 )( p μ 1 ) 1 B z, 7.6 {( Jδ,p f ) z } Γ ( p 1 ) ( Γ ( cp ( δ p ) A B ( p λ ) ) p μ 1 ) z pμ 1 a ( p α )( p δ 1 )( p μ 1 ) 1 B z 7.7 μ>0; δ> p; p N; z U, each of the assertions is sharp. Proof. Since D μ {( z Jδ,p f ) z } Γ ( p 1 ) Γ ( p μ 1 )zpμ ( ) ( ) δ p Γ k p 1 ( ) ( ) akpz kpμ, 7.8 δ k p Γ k p μ 1 then D μ z {( Jδ,p f ) z } Γ ( p 1 ) ( ) ( ) ( ) Γ ( δ p Γ k p 1 Γ p μ 1 p μ 1 ) zμ zp ( ) ( ) ( ) akpz kp δ k p Γ k p μ 1 Γ p Let Gz is defined by ( ) ( ) ( ) δ p Γ k p 1 Γ p μ 1 Gz z p ( ) ( ) ( ) akpz kp z p Qk akpz kp, δ k p Γ k p μ 1 Γ p where ( δ p ) Γ ( k p 1 ) Γ ( p μ 1 ) Qk ( ) ( ) ( ) δ k p Γ k p μ 1 Γ p 1 ( k, p N; μ>0 ) Since Qk is a decreasing function of k when μ>0, we get ( )( ) δ p p 1 0 <Qk Q1 ( )( ), 7.12 δ k p 1 p μ

18 18 International Journal of Mathematics and Mathematical Sciences since fz Y a,ca, B; p, λ, α, bytheorem 2.1, weget a ( p α ) 1 B ( p 1 ) ( )( ) cpa B ( p λ ) p α k p 1 Bak akp pa B ( p λ ) akp c k It is that cpa B ( p λ ) akp a ( α p ) 1 B ( p 1 ), 7.14 then Gz zp Qk a kpz kp z p Q1 z p1 a kp z p cp ( p δ ) A B ( p λ ) a ( p α )( δ k p )( p 1 μ ) 1 B z p1, Gz zp Qk a kpz kp z p Q1 z p1 akp z p cp ( p δ ) A B ( p λ ) a ( p α )( δ k p )( p 1 μ ) 1 B z p From 7.15, weobtain7.6 and 7.7, respectively. Equalities in 7.6 and 7.7 are attained for the function fz given by D μ {( z Jδ,p f ) z } Γ ( p 1 ) ( Γ ( p μ 1 )zpμ 1 cp ( δ p ) A B ( p λ ) ) a ( p α )( δ k p )( p μ 1 ) 1 B z Theorem 7.6. Let function defined by 1.1 be in the class Y a,ca, B; p, λ, α, then D μ {( z Jδ,p f ) z } Γ ( p 1 ) ( Γ ( cp ( δ p ) A B ( p λ ) ) p μ 1 ) z p μ 1 a ( p α )( p δ 1 )( p μ 1 ) 1 B z, D μ {( z Jδ,p f ) z } Γ ( p 1 ) ( Γ ( p μ 1 ) z p μ 1 cp ( δ p ) A B ( p λ ) a ( p α )( p δ 1 )( p μ 1 ) 1 B z 7.17 ), 7.18 μ>0; δ> p; p N; z U, each of the assertions is sharp.

19 International Journal of Mathematics and Mathematical Sciences 19 Proof. Since D μ z{( Jδ,p f ) z } Γ ( p 1 ) Γ ( p μ 1 )zp μ ( ) ( ) δ p Γ k p 1 ( ) ( ) akpz kp μ, 7.19 δ k p Γ k p μ 1 then D μ {( z Jδ,p f ) z } Γ ( p 1 ) Γ ( p μ 1 ) z μ zp ( ) ( ) ( ) δ p Γ k p 1 Γ p μ 1 ( ) ( ) ( ) akpz kp δ k p Γ k p μ 1 Γ p Let Hz be defined by ( ) ( ) ( ) δ p Γ k p 1 Γ p μ 1 Hz z p ( ) ( ) ( ) akpz kp δ k p Γ k p μ 1 Γ p 1 z p Ek ( k p ) akpz kp, 7.21 where ( δ p ) Γ ( k p ) Γ ( p μ 1 ) Ek ( ) ( ) ( ) δ k p Γ k p μ 1 Γ p 1 ( k, p N; μ>0 ) Since Ek is a decreasing function of k when 0 μ<1, then ( ) δ p 0 <Ek E1 ( )( ), 7.23 δ 1 p 1 p μ since fz Y a,ca, B; p, λ, α, bytheorem 2.1, weget a ( p α ) 1 B cpa B ( p λ ) ( k p ) akp ( p α )( k p ) 1 Bak pa B ( p λ ) c k akp It is that ( ) k p a kp cpa B( p λ ) a ( α p ) 1 B, 7.25

20 20 International Journal of Mathematics and Mathematical Sciences then Hz zp Ek ( k p ) akpz kp z p E1 z p1 ( ) k p akp z p cp ( p δ ) A B ( p λ ) a ( p α )( δ k p )( p 1 μ ) 1 B z p1, Hz zp Ek ( k p ) akpz kp z p E1 z p1 ( ) k p akp z p cp ( p δ ) A B ( p λ ) a ( p α )( δ k p )( p 1 μ ) 1 B z p From 7.26, weobtain7.17 and 7.18, respectively. Equalities in 7.17 and 7.18 are attained for the function fz given by Dz{( μ Jδ,p f ) z } Γ ( p 1 ) ( Γ ( cp ( δ p ) A B ( p λ ) ) p μ 1 )zp μ 1 a ( p α )( δ k p )( p μ 1 ) 1 B z 7.27 or equivalently by ( Jδ,p f ) z z p cp ( δ p ) A B ( p λ ) a ( p α )( δ k p )( p μ 1 ) 1 B zp Acknowledgments The author thanks the referee for numerous suggestions that helped make this paper more readable. The present investigation was partly supported by the Natural Science Foundation of Inner Mongolia under Grant 2009MS0113. References 1 M. K. Aouf, H. M. Hossen, and H. M. Srivastava, Some families of multivalent functions, Computers & Mathematics with Applications, vol. 39, no. 7-8, pp , S. Owa, Some properties of certain multivalent functions, Applied Mathematics Letters, vol. 4, no. 5, pp , H. Saitoh, A linear operator and its applications of first order differential subordinations, Mathematica Japonica, vol. 44, no. 1, pp , H. M. Srivastava and J. Patel, Some subclasses of multivalent functions involving a certain linear operator, Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp , V. Kumar and S. L. Shukla, Multivalent functions defined by Ruscheweyh derivatives. I, II, Indian Journal of Pure and Applied Mathematics, vol. 15, no. 11, pp , M. K. Aouf, H. Silverman, and H. M. Srivastava, Some families of linear operators associated with certain subclasses of multivalent functions, Computers & Mathematics with Applications,vol.55,no.3, pp , 2008.

21 International Journal of Mathematics and Mathematical Sciences 21 7 J. Sokół, Classes of multivalent functions associated with a convolution operator, Computers & Mathematics with Applications, vol. 60, no. 5, pp , M. K. Auof and H. E. Darwish, Some classes of multivalent functions with negative coefficients. I, Honam Mathematical Journal, vol. 16, no. 1, pp , S. L. Shukla and Dashrath, On certain classes of multivalent functions with negative coefficients, Soochow Journal of Mathematics, vol. 8, pp , S. K. Lee, S. Owa, and H. M. Srivastava, Basic properties and characterizations of a certain class of analytic functions with negative coefficients, Utilitas Mathematica, vol. 36, pp , V. P. Gupta and P. K. Jain, Certain classes of univalent functions with negative coefficients. II, Bulletin of the Australian Mathematical Society, vol. 15, no. 3, pp , B. A. Uralegaddi and S. M. Sarangi, Some classes of univalent functions with negative coefficients, Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi, vol. 34, no. 1, pp. 7 11, O. Altintas and S. Owa, Neighborhoods of certain analytic functions with negative coefficients, International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 4, pp , O. Altintaş, Ö. Özkan, and H. M. Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Applied Mathematics Letters, vol. 13, no. 3, pp , O. Altintaş, Ö. Özkan, and H. M. Srivastava, Neighborhoods of a certain family of multivalent functions with negative coefficients, Computers & Mathematics with Applications, vol. 47, no , pp , M. K. Aouf, Neighborhoods of certain classes of analytic functions with negative coefficients, International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 38258, 6 pages, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics, and Its Applications, Horwood, Chichester, UK. 18 M.-P. Chen, H. Irmak, and H. M. Srivastava, Some families of multivalently analytic functions with negative coefficients, Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp , H. M. Srivastava and M. K. Aouf, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. II, Journal of Mathematical Analysis and Applications, vol. 192, no. 3, pp , R. J. Libera, Some classes of regular univalent functions, Proceedings of the American Mathematical Society, vol. 16, pp , A. E. Livingston, On the radius of univalence of certain analytic functions, Proceedings of the American Mathematical Society, vol. 17, pp , H. M. Srivastava and S. Owa, Eds., Current Topics in Analytic Function Theory, World Scientific Publishing, River Edge, NJ, USA, S. Owa, On distortion theorem, I, Kyungpook Mathematical Journal, vol. 18, pp , 1978.

22 Advances in Operations Research Advances in Decision Sciences Mathematical Problems in Engineering Journal of Algebra Probability and Statistics The Scientific World Journal International Journal of Differential Equations Submit your manuscripts at International Journal of Advances in Combinatorics Mathematical Physics Journal of Complex Analysis International Journal of Mathematics and Mathematical Sciences Journal of Stochastic Analysis Abstract and Applied Analysis International Journal of Mathematics Discrete Dynamics in Nature and Society Journal of Journal of Discrete Mathematics Journal of Applied Mathematics Journal of Function Spaces Optimization