Certain classes of p-valent analytic functions with negative coefficients and (λ, p)-starlike with respect to certain points

Transcription

1 BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 379), 2015, Pages ISSN Certain classes of p-valent analytic functions with negative coefficients and λ, p)-starlike with respect to certain points Adnan Ghazy Alamoush, Maslina Darus Abstract. In this article, we consider classes Ss,λp, α, β), Sc,λp, α, β), and Ssc,λp,α, β) of p-valent analytic functions with negative coefficients in the unit disk. They are, respectively, λ, p)-starlike with respect to symmetric points, λ, p)-starlike with respect to conjugate points, and λ, p)-starlike with respect to symmetric conjugate points. Necessary and sufficient coefficient conditions for functions f belonging to these classes are obtained. Several properties such as the coefficient estimates, growth and distortion theorems, extreme points, radii of starlikeness, convexity, and integral operator are studied. Mathematics subject classification: 30C45. Keywords and phrases: p-valent functions, univalent functions, Salagean operator, starlike with respect to symmetric points. 1 Introduction Let A denote the class of functions fz) of the form fz) = z + a k z k, 1) which are analytic in the punctured unit disk U = {z : z < 1}. For f which belong to A, Salagean [1] introduced the following operator: k=2 D 0 fz) = fz), D 1 fz) = Dfz) = zf z), and D n fz) = D D n 1 fz) ) n N = {1,2,3,...}). 2) Note that D n fz) = D D n 1 fz) ) = z + k n a k z k n N 0 = {0} N). 3) k=2 Let T p p a fixed integer greater than 0) denote the class of functions of the form fz) = z p + a k+p z k+p 4) c Adnan Ghazy Alamoush, Maslina Darus,

2 36 ADNAN GHAZY ALAMOUSH, MASLINA DARUS that are holomorphic and p-valent in the unit disk z < 1. Also let T p denote the subclass of S p consisting of functions that can be expressed in the form fz) = z p a k+p z k+p 5) where a k+p 0, p N = {1,2,3,...}, n N, which are analytic in the unit disc U. We can write the following equalities for the functions fz) which belong to the class T p see [2]): D 0 fz) = fz), D 1 fz) = Dfz) = zf z) ) = z pz p k + p)a k+p z k+p 1 = pz p k + p)a k+p z k+p, D 2 fz) = DDfz)) = p 2 z p... k + p) 2 a k+p z k+p, ) D λ fz) = D D λ 1 fz) = p λ z p k + p) λ a k+p z k+p. 6) Let S be the subclass of A consisting of functions that are regular and univalent in U. Let S be the subclass of S consisting of functions starlike in U. It is known that } f S if and only if R > 0, z U). { zf z) fz) In [3], Sakaguchi defined the class of starlike functions with respect to symmetric points as follows: Let f S. Then f is said to be starlike with respect to symmetric points in U if and only if { zf } z) R > 0, z U), 7) fz) f z) and we denote this class by Ss. Obviously, it forms a subclass of close-to-convex functions and hence univalent. Moreover, this class includes the class of convex functions and odd starlike functions with respect to the origin, see Sakaguchi [3], Robertson [5], Stankiewicz [6], Wu [7] and Owa et al. [8]. El-Ashwah and Thomas in [9] introduced two other classes, namely the class Sc consisting of functions starlike with respect to conjugate points and Ssc consisting of functions starlike with

3 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS respect to symmetric conjugate points. The class Ssc [10] see also [11]). is also studied by Chen et al. In [4], Sudharsan et al. introduced the class Ssα,β) of functions fz) S and satisfying the following condition see also [12]): zf z) fz) f z) 1 < β zf z) α fz) f z) + 1 8) for some 0 α 1, 0 < β 1, z U. Recently, Aouf el at.[13] introduced the class Ss,n T α,β) of functions fz) being defined by 5). Then fz) is said to be n-starlike with respect to symmetric points if it satisfies the following condition: D n+1 fz) D n fz) D n f z) 1 < β D n+1 fz) α D n fz) D n f z) + 1, 9) where n N 0 = {0} N,0 α 1, 0 < β 1, 0 21 β) 1+αβ, and z U. However, in this paper we consider the subclass T defined by 5). Definition 1. Let a function fz) be defined by 5). Then fz) is said to be λ,p)- starlike with respect to symmetric points if it satisfies the following condition: D λ+1 fz) D λ fz) D λ f z) p < β D λ+1 fz) α D λ fz) D λ f z) + p, 10) where λ N 0 = {0} N, p N, 0 α 1, 0 < β 1, 0 2p1 β) 1+αβ, and z U. We denote the class of functions λ,p)-starlike with respect to symmetric points by Ss,λ p,α,β). Definition 2. Let a function fz) be defined by 5). Then fz) is said to be λ,p)- starlike with respect to conjugate points if it satisfies the following condition: D λ+1 fz) D λ fz) + D λ fz) p < β α D λ+1 fz) D λ fz) + D λ fz) + p, 11) where λ N 0 = {0} N, p N, 0 α 1, 0 < β 1, 0 2p1 β) 1+αβ, and z U. We denote the class of functions λ, p)-starlike with respect to conjugate points by Sc,λ p,α,β). Definition 3. Let a function fz) be defined by 5). Then fz) is said to be λ, p)-starlike with respect to symmetric conjugate points if it satisfies the following condition: D λ+1 fz) D λ fz) D λ f z) p < β α D λ+1 fz) D λ fz) D λ f z) + p, 12)

4 38 ADNAN GHAZY ALAMOUSH, MASLINA DARUS where λ N 0 = {0} N, p N, 0 α 1, 0 < β 1, 0 2p1 β) 1+αβ, and z U. We denote the class of functions λ, p)-starlike with respect to symmetric conjugate points by Ssc,λ p,α,β). Notice that the above conditions imposed on α, β and p in the introduction are necessary to ensure that these classes form a subclass of S. For more classes see in details Halim et al. [14,15]. 2 Coefficient estimates To prove the following theorems, we will adopt the technique used by Dziok [16], and assume that λ N 0 = {0} N, p N, 0 α 1, 0 < β 1, 0 2p1 β) 1+αβ, and z U. Theorem 1. Let the function fz) be defined by 5) and D λ fz) D λ f z) 0 for z 0. Then fz) Ss,λ p,α,β) if and only if [ )] k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) k + p) λ+1 a k+p Proof. Use 5), 6) and 10), that is and + p λ+1 [ 1) p + βα + 1 1) p )]. 13) D λ+1 fz) D λ fz) D λ f z) p < β D λ+1 fz) α D λ fz) D λ f z) + p [ = 1)p p λ+1 z p k + p 1) p+k] k + p) λ a k+p z k+p < β α + 1 1)p )p λ+1 z p [αk + pα + 1 1) p+k )] a k+p z k+p p λ+1 [ 1) p + βα + 1 1) p )] z p [ ] k + p 1) p+k + β{αk + pα + 1 1) p+k )} k + p) λ a k+p z k+p 0. Letting z = 1, we have [ ] k + p 1) p+k + β{αk + pα + 1 1) p+k )} k + p) λ a k+p 0. p λ+1 [ 1) p + βα + 1 1) p )] 0.

5 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS Therefore, by the maximum modulus theorem, we have fz) S s,λ p,α,β). For the converse, let us suppose that D λ+1 fz) D λ fz) D λ f z) p D α λ+1 fz) < β. D λ fz) D λ f z) + p This implies that [ 1) p p λ+1 z p + [ k + p 1) p+k ] k + p) λ a k+p z k+p] α + 1 1) p )p λ+1 z p [αk + pα + 1 1)p+k )] a k+p z k+p < β. Since Rz) z for all z, we have { 1) p p λ+1 z p + [ k + p 1) p+k ] } k + p) λ a k+p z k+p R α + 1 1) p )p λ+1 z p [αk + pα + 1 1)p+k )] a k+p z k+p < β. 14) D If we choose values of z on the real axis, then λ+1 fz) D λ fz) D λ f z) is real and Dλ fz) D λ f z) 0 for z 0. Upon clearing the denominator in 14) and letting z 1 through real values, we obtain [ k + p 1) p+k] k + p) λ a k+p + β[αk + pα + 1 1) p+k )]k + p) λ a k+p This gives the required condition. βα + 1 1) p )p λ+1 + 1) p p λ+1. Corollary 1. Let the function fz) defined by 5) be in the class Ss,λ p,α,β). Then we have a k+p p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p 1) p+k + βα + 1 1) p+k k 1), 15) ))] k + p) λ+1 where p N, λ N 0 and z U. The equality in 15) is attained for the function fz) given by f k z) = z p p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p 1) p+k + βα + 1 1) p+k ))]k + p) λ+1zk+p k 1), where p N, λ N 0 and z U. Theorem 2. Let the function fz) be defined by 5) and D λ fz) D λ f z) 0 for z 0. Then fz) Sc,λ p,α,β) if and only if [k1 + αβ) + p βα + 2) 1)] k + p) λ+1 a k+p 16) p λ+1 [βα + 2) 1)]. 17)

6 40 ADNAN GHAZY ALAMOUSH, MASLINA DARUS Corollary 2. Let the function fz) defined by 5) be in the class Sc,λ p,α,β). Then we have p λ+1 [βα + 2) 1)] a k+p [k1 + αβ) + βα + 2) 1)]k + p) λ+1 k 1), 18) where p N, λ N 0 and z U. The equality in 18) is attained for the function fz) given by f k z) = z p where p N, λ N 0 and z U. p λ+1 [βα + 2) 1)] [k1 + αβ) + βα + 2) 1)] k + p) λ+1zk+p k 1), 19) Theorem 3. Let the function fz) be defined by 5) and D λ fz) D λ f z) 0 for z 0. Then fz) Ssc,λ p,α,β) if and only if [ k1 + αβ) + p β1 + α) + 1 β) 1) p+k)] k + p) λ+1 a k+p p λ+1 [ 1) p + βα + 1 1) p )]. 20) Corollary 3. Let the function fz) defined by 5) be in the class Ssc,λ p,α,β). Then we have a k+p p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p β1 + α) + 1 β) 1) p+k )]k + p) λ+1 k 1), 21) where p N, λ N 0 and z U. The equality in 21) is attained for the function fz) given by f k z) = z p p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p β1 + α) + 1 β) 1) p+k )]k + p) λ+1zk+p k 1), where p N, λ N 0 and z U. 3 Growth and Distortion theorems Theorem 4. Let the function fz) defined by 5) be in the class fz) S s,λ p,α,β). Then we have p i z p p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 z p+1 ))]p + 1) λ i D i fz) 22) p i z p p λ+1 [ 1) p + βα + 1 1) p )] + [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))]p + 1) λ i z p+1 23) for z U, where 0 i n. The result is sharp.

7 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS Proof. Note that fz) Ss,λ p,α,β) if and only if Di fz) Ss,λ i p,α,β), and that D i fz) = p i z p Using Theorem 1, we know that k + p) i a k+p That is k+p) i a k+p It follows from 24) and 26) that p i z p k + p) i a k+p z k+p. 24) p λ+1 [ 1) p + βα + 1 1) p )] p + 1) i [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))] p + 1) λ. 25) p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))]p + 1) λ i. 26) D i fz) p i z p z p+1 k + p) i a k+p p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+2 ))]p + 1) λ i z p+1. 27) Also D i fz) p i z p + z p+1 k + p) i a k+p z k+p p i z p + p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))]p + 1) λ i z p+1. 28) Finally, we note that the equality in 23) is attained by the function D i fz) = p i z p or by fz) = z p p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))]p + 1) λ izp+1 29) p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))]p + 1) λzp+1 30) and this completes the proof of Theorem 4.

8 42 ADNAN GHAZY ALAMOUSH, MASLINA DARUS Corollary 4. Let the function fz) defined by 5) be in the class fz) S s,λ p,α,β). Then we have z p p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))] p + 1) λ z p+1 fz) z p p λ+1 [ 1) p + βα + 1 1) p )] + [1 + αβ) + p 1) p+1 + βα + 1 1) p+1 ))] p + 1) λ z p+1 31) for z U. The result is sharp for the function fz) given by 30). Proof. For i = 0 in Theorem 4, we can easily show 30). Similarly, we can prove the following result. Theorem 5. Let the function fz) defined by 5) be in the class fz) S c,λ p,α,β). Then we have p i z p p λ+1 [βα + 2) 1)] z p+1 [1 + αβ) + p βα + 2) 1)] p + 1) λ i D i fz) p i z p p λ+1 [βα + 2) 1)] + [1 + αβ) + p βα + 1) 1)] p + 1) λ i z p+1 32) for z U, where 0 i n. The result is sharp, for the function fz) given by or by D i fz) = p i z p fz) = z p p λ+1 [βα + 2) 1)] [1 + αβ) + p βα + 2) 1)] p + 1) λ izp+1 33) p λ+1 [βα + 2) 1)] [1 + αβ) + p βα + 2) 1)] p + 1) λzp+1. 34) Corollary 5. Let the function fz) defined by 5) be in the class fz) S c,λ p,α,β). Then we have z p p λ+1 [βα + 2) 1)] [1 + αβ) + p βα + 2) 1)]p + 1) λ z p+1 fz) z p p λ+1 [βα + 2) 1)] + [1 + αβ) + p βα + 2) 1)]p + 1) λ z p+1 35) for z U. The result is sharp, for the function fz) given by 34).

9 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS Theorem 6. Let the function fz) defined by 5) be in the class fz) S sc,λ p,α,β). Then we have p i z p p λ+1 [ 1) p + βα + 1 1) p )] [1 + αβ) + p β1 + α) + 1 β) 1) p+1 z p+1 )]p + 1) λ i D i fz) p i z p p λ+1 [ 1) p + βα + 1 1) p )] + [1 + αβ) + p β1 + α) + 1 β) 1) p+1 )]p + 1) λ i z p+1 36) for z U, where 0 i n. The result is sharp. 4 Extreme points Theorem 7. Let f p z) = z p and f k+p z) = z p p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p 1) p+k + βα + 1 1) p+k ))]k + p) λ+1zk+p, 37) where k 1. Then fz) Ss,λ p,α,β) if and only if it can be expressed in the form fz) = σ k+p f k+p z), 38) k=0 where σ k+p 0 k 1) and k=0 σ k+p = 1. Proof. Suppose = z p fz) = σ k+p f k+p z) k=0 p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p 1) p+k + βα + 1 1) p+k ))]k + p) λ+1σ k+pz k+p. Then we get [ k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 39) p λ+1 [ 1) p + βα + 1 1) p )] p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p 1) p+k + βα + 1 1) p+k ))]k + p) λ+1σ k+p = σ k+p = 1 σ p 1. 40)

10 44 ADNAN GHAZY ALAMOUSH, MASLINA DARUS It follows from Theorem 1 that the function fz) S s,λ p,α,β). Conversely, suppose that fz) Ss,λ p,α,β). Again, by using Theorem 1, we can show that a p+k p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p 1) p+k + βα + 1 1) p+k k 1, 41) ))]k + p) λ+1 where k 1, λ N 0 = {0} N, p N, 0 α 1, 0 < β 1 and z U. Setting σ p+k and [ k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 p λ+1 [ 1) p + βα + 1 1) p )] σ p = 1 k 1) 42) σ k+p, 43) we can see that fz) can be expressed in the form 38). This completes the proof of Theorem 7. Corollary 6. The extreme points of the class S s,λ p,α,β) are functions f k+pz) k 1, p N) given by Theorem 7. Similar to Theorem 7, we can easily prove the following theorems for Sc,λ p,α,β) and Ssc,λ p,α,β) classes. Theorem 8. Let f p z) = z p and f k+p z) = z p p λ+1 [βα + 2) 1)] [k1 + αβ) + βα + 2) 1)]k + p) λ+1zk+p k 1), 44) where p N, λ N 0 and z U. Then fz) Sc,λ p,α,β) if and only if it can be expressed in the form fz) = σ k+pf k+p z), where σ k+p 0 k 1) and σ k+p = 1. Corollary 7. The extreme points of the class S c,λ p,α,β) are functions f k+pz) k 1, p N) given by Theorem 8. Theorem 9. Let f p z) = z p and f k+p z) = z p p λ+1 [ 1) p + βα + 1 1) p )] [k1 + αβ) + p β1 + α) + 1 β) 1) p+k )] k + p) λ+1zk+p k 1), 45) where p N, λ N 0 and z U. Then fz) Ssc,λ p,α,β) if and only if it can be expressed in the form fz) = σ k+pf k+p z), where σ k+p 0 k 1) and σ k+p = 1.

11 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS Corollary 8. The extreme points of the class S sc,λ p,α,β) are functions f k+pz) k 1, p N) given by Theorem 9. Theorem 10. The class Ss,λ p,α,β) is closed under convex linear combination. Proof. Let us suppose that the functions f 1 z) and f 2 z) defined by f j z) = z p a p+k,j z p+k a p+k,j 0, j = 1,2, z U) 46) are in the class Ss,λ p,α,β). Set Then from 46), we can write hz) = µf 1 z) + 1 µf 2 z)), 0 µ 1). 47) fz) = z p [µ a p+k,1 + 1 µ) a p+k,1 ]z p+k a p+k,j 0, j = 1,2, z U). 48) k=2 Thus, in view of Theorem 1, we can have that k=2 [ )] k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) k + p) λ+1 a k+p [µ a p+k,1 ] +1 µ) a p+k,1 p λ+1 [ 1) p + βα + 1 1) p )], which implies that hz) Ss,λ p,α,β) and this completes the proof of Theorem Radii of starlikeness and Convexity Theorem 11. Let the function fz) of the form 5) be in the class S s,λ p,α,β), then fz) is starlike in the disk z = r 1 < 1, where r 1 = inf k 1 [ k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 Proof. It is ample to show that p λ+1 [ 1) p + βα + 1 1) p )] zf z) fz) p p, for z < 1 ) ) 1 p k. p + k 49) or equivalently, k a k+p z k 1 a k+p z k p

12 46 ADNAN GHAZY ALAMOUSH, MASLINA DARUS which is equivalent to show that k + 1) a k+p z k As fz) Ss,λ p,α,β) we have from Theorem 1 p 1. 50) [ k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 p λ+1 [ 1) p + βα + 1 1) p a k+p 1. )] Hence, 50) is proven true if ) [ p + k) z k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 p p λ+1 [ 1) p + βα + 1 1) p. )] That is, r 1 = inf k 1 [ k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 and this ends the proof of Theorem 11. p λ+1 [ 1) p + βα + 1 1) p )] p ) ) 1 k p + k On similar lines of Theorem 11, we can easily prove the following Theorems for Sc,λ p,α,β) and S sc,λ p,α,β) classes. Theorem 12. Let the function fz) of the form 5) be in the class S c,λ p,α,β), then fz) is starlike in the disk z = r 1 < 1, where )) [k1 + αβ) + p βα + 2) 1)] k + p) λ+1 p 1 k r 1 = inf. 51) k 1 p λ+1 [βα + 2) 1)] p + k Theorem 13. Let the function fz) of the form 5) be in the class S sc,λ p,α,β), then fz) is starlike in the disk z = r 1 < 1, where r 1 = inf k 1 [ k1 + αβ) + p β1 + α) + 1 β) 1) p+k )] k + p) λ+1 p λ+1 [ 1) p + βα + 1 1) p )] Similarly we can proved the following Results. ) ) 1 p k. p + k 52) Theorem 14. Let the function fz) of the form 5) be in the class S s,λ p,α,β), then fz) is convex in the disk z = r 2 < 1, where r 1 = inf k 1 [ k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) )] k + p) λ+1 p λ+1 [ 1) p + βα + 1 1) p )] p p + k ) 2 ) 1 k. 53)

13 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS Theorem 15. Let the function fz) of the form 5) be in the class S c,λ p,α,β), then fz) is convex in the disk z = r 2 < 1, where r 1 = inf k 1 [k1 + αβ) + p βα + 2) 1)]k + p) λ+1 p λ+1 [βα + 2) 1)] ) ) 1 p 2 k. 54) p + k Theorem 16. Let the function fz) of the form 5) be in the class S sc,λ p,α,β), then fz) is convex in the disk z = r 2 < 1, where r 1 = inf k 1 [ k1 + αβ) + p β1 + α) + 1 β) 1) p+k )] k + p) λ+1 p λ+1 [ 1) p + βα + 1 1) p )] ) ) 1 p 2 k. p + k 55) In order to establish the required results in Theorems 14, 15 and 16, it is sufficies to show that 1 + zf z) f z) p p, for z < 1. 6 Integral Operator Definition 4. Let f T p, an integral operator R c f) with c > p is defined by R c f) = c + p z c z 0 t c 1 ft)dt, z U). 56) Theorem 17. Let the function fz) of the form 5) be in the class S s,λ p,α,β), then R c f) defined by 56) be also in the class S s,λ p,α,β). Proof. From 56), we get R c f) = z p c + p c + p + k a k+p z k+p. 57) Therefore by hypothesis [ )] ) c + p k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) k + p) λ+1 a k+p c + p + k [ )] k1 + αβ) + p 1) p+k + βα + 1 1) p+k ) k + p) λ+1 a k+p p λ+1 [ 1) p + βα + 1 1) p )] since fz) S s,λ p,α,β). Hence, by Theorem 1, R cf) S s,λ p,α,β). Similar to Theorem 17, we can easily prove the following theorems for S c,λ p,α,β) and S sc,λ p,α,β).

14 48 ADNAN GHAZY ALAMOUSH, MASLINA DARUS Theorem 18. Let the function fz) of the form 5) be in the class S c,λ p,α,β), then R c f) defined by 56) be also in the class S c,λ p,α,β). Theorem 19. Let the function fz) of the form 5) be in the class S sc,λ p,α,β), then R c f) defined by 56) be also in the class S sc,λ p,α,β). Acknowledgements. The authors would like to acknowledge and appreciate the financial support received from Universiti Kebangsaan MalaysiaUKM) under the grant: AP The authors also would like to thank the referee for the comments given to improve the manuscript. References [1] Salagean G. S. Subclasses of univalent functions. Lecture Notes Math., 1013, Springer Verlag, Berlin, 1983, [2] Orhan H., Kiziltunc H. A generalization on subfamily of p-valent functions with negative coefficients. Appl. Math. Comp., 2004, 155, [3] Sakaguchi K. On certain univalent mapping. J. Math. Soc. Japan, 1959, 11, [4] Sudharsan T. V., Balasubrahmananyam P., Subramanian K. G. On functions starlike with respect to symmetric and conjugate points, Taiwanese J. Math., 1998, 2, [5] Robertson M.S. Applications of the subordination principle to univalent functions. Pacific J. Math., 1961, 11, [6] Stankiewicz J. Some remarks on functions starlike with respect to symmetric points. Ann. Univ. Marie Curie Sklodowska, 1965, 19, [7] Wu Z. On classes of Sakaguchi functions and Hadamard products. Sci. Sinica Ser., 1987, A 30, [8] Owa S., Wu Z., Ren F. A note on certain subclass of Sakaguchi functions. Bull. Soc. Roy. Liege, 1988, 57, [9] El-Ashwah R. M., Thomas D. K. Some subclasses of close-to-convex functions. J. Ramanujan Math. Soc., 1987, 2, [10] Chen M.-P., Wu Z.-R., Zou Z.-Z. On functions α-starlike with respect to symmetric conjugate points. Jour. Math. Anal. App. 201, 1996, Art. No. 0238, [11] Sokól J. Function starlike with respect to conjugate point. Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz., 1991, 12, [12] Sokól J. Some remarks on the class of functions starlike with respect to symmetric points. Folia Scient. Univ. Tech. Resoviensis, 1990, 73, [13] Aouf M. K., El-Ashwah R. M., El-Deeb S. M. Certain classes of univalent functions with negative coefficients and n-starlike with respect to certain points. Matematiqki Vesnik, 2010, 62, No. 3, [14] Halim S. A., Janteng A., Darus M. Coefficient properties for classes with negative coefficient and starlike with respect to other points. Proceeding of The 13th Math. Sci. Nat. Symposium, UUM., 2005, 2,

15 CERTAIN CLASSES OF P-VALENT ANALYTIC FUNCTIONS [15] Halim S. A., Janteng A., Darus M. Classes with negative coefficient and starlike with respect to other points. Int. Math., Forum 2, 2007, 46, [16] Dziok P. On the convex combination of the Dziok-Srivastava operator. Appl. Math. Comp., 2006, 188, Adnan Ghazy Alamoush, Maslina Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia UKM Bangi Selangor, Malaysia adnan Received May 10, 2015