Certain Subclass of p Valent Starlike and Convex Uniformly Functions Defined by Convolution

Int. J. Oen Problems Comt. Math., Vol. 9, No. 1, March 2016 ISSN 1998-6262; Coyright c ICSRS Publication, 2016 www.i-csrs.org Certain Subclass of Valent Starlie and Convex Uniformly Functions Defined by Convolution M. K. Aouf, A. O. Mostafa and A. A. Hussain Deartment of Mathematics, Faculty of Science Mansoura University, Mansoura, Egyt e-mail: maouf127@yahoo.com e-mail: adelaeg254@yahoo.com e-mail: aisha84 hussain@yahoo.com Received 2 January 2016; Acceted 1 March 2016 Abstract The aim of this aer is to obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to convexity, starlienees and convexity for functions belonging to the subclass T S,λ f, g; α, β of -valent β uniformaly starlie and convex functions. We consider integral oerators and modified Hadamarad roducts of functions belonging to this subclass. Keywords: Analytic, -valent, uniformly starlie, uniformly convex, modified Hadamard roduct. 2000 Mathematical Subject Classification: 30C45. 1 Introduction The class of analytic and -valent in U and has the form: fz = z + a z N, N = {1, 2,...} 1 =+1

Certain Subclass of Starlie and Convex Uniformly Functions 37 is denoted by S. We note that S1 = S. Let fz S be given by 1 and gz S be given by gz = z + =+1 b z, 2 then the Hadamard roduct f gz of fz and gz is defined by f gz = z + a b z = g fz. 3 =+1 Denote by T the class of functions of the form: fz = z a z a 0. 4 =+1 Salim et al. [20] and Marouf [16] introduced and studied the following subclasses of valent functions: i A function fz S is said to be valent β uniformly starlie of order α if it satisfies the condition: { } zf z Re fz α β zf z fz z U, 5 for some α α <, β 0 and for all z U. denote to the class of these functions by β S α. ii A function fz S is said to be valent β uniformly convex of order α if it satisfies the condition: Re {1 + } zf z f z α β zf z 1 + f z z U, 6 for some α α <, β 0 and for all z U. We denote to the class of these functions by β K α. It follows from 5 and 6 that fz β K α zf β S α. 7 For different choices of arameters α, β, we obtain many subclasses studied earlier see for examle [1], [11], [12], [13], [14], [18] and [19] Definition 1. For 0 α <, N, 0 λ 1 and β 0, let S,λ f, g; α, β be the subclass of S consisting of functions fz of the form 1 and functions gz of the form 2 and satisfying the analytic criterion: { 1 λ + λ Re zf } z + λ g z2 f g z 1 λ f gz + λzf α 8 z g

38 M. K. Aouf, A. O. Mostafa and A. A. Hussain and 1 λ + λ > β zf g z + λ z2 f g z 1 λ f gz + λzf z U 9 g z T S,λ f, g; α, β = S,λ f, g; α, β T. 10 We note that: it S 1,λ f, g; α, β = T S γ f, g; α, β see Aouf et al. [3]; iit S,0 f, z + Ω z ; α, β = T l,m α l, β m, β, αsee Marouf [16], where and a = Γa + Γa =+1 = Ω = α 1...α l β 1...β m 1! 11 { 1 = 0 aa + 1a + 2...a + 1 N, 12 0 α <, β 0, α i Ci = 1, 2,...l and β j C\{ 1, 2,...}, j = 1, 2,...m, z U; iiit S,0 f, z + al.[20], where =+1 φ n, γ, δ, = φ n, γ, δ, z ; α, β = S n α, β, γ, δsee Salim et [ 1 γ 1 + ] 1δ + γcn+ n, 13 n N 0 = N {0}, γ 0, δ 0, α <, β 0 14 and Cn+ n n +! =. 15 n!! ivt S,0 f, z + C, n1 γ n z + = Sα, β, γ and T S,1 f, z + =1 C, n1 γ n z + = C α, β, γsee Darus and Ibrahim [7, by relacing =1 by ], where N, n N 0, 0 γ < 1, 0 α <, β 0, z U; vit S,0 f, z + =+n [15, with n = 1], where Φ = C, n = n + 1 16 Φz = Kµ, γ, η, α, βsee Khairnar and More a 1 + 1 + + η γ c 1 + γ 1 + + η µ 17

Certain Subclass of Starlie and Convex Uniformly Functions 39 < µ < 1, < γ < 1, η R +, N, α <, β 0, a R, c R\Z 0, z U. Also we note that: it S,λ f, z + n z ; α, β = T S,λ n, α, β =+1 f T : Re = > β { 1 λ+ λ zdn fz + λ z2 D n fz 1 λd n λ, fz+ λ zdn fz 1 λ+ λ zdn fz + λ z2 D n fz 1 λd n λ, fz+ λ zdn fz } α 18 0 λ 1, α < ; β 0, n N 0, z U, where D n is valent Sãlãgean oerator introduced by [10] and [4]; Ω z ; α, β = T S,n λ,q,s n, α, β =+1 = { } 1 λ+ f T : Re λ zh,q,sα 1fz + λ z2 H,q,sα 1 fz α 1 λh,q,sα 1 fz+ λ zh,q,sα 1z > β 1 λ+ λ zh,q,sα 1fz + λ z2 H,q,sα 1 fz 1 λh,q,sα 1 fz+ λ zh,q,sα 1z z U, 19 where Ω is given by 1.10 and H,q,s α 1 is the valent Dzio-Srivastava oeratorintroduced by [8] and [9], see also [5], [2] and [6]. Denote by U the unit disc of... Remar 1.1 If f A ζ, fz, ζ = z + j=2 a j ζ z j, then R m f z, ζ = z + j=2 Cm m+j 1a j ζ z j, z U, ζ U. 2 Coefficient estimates Throughout our resent aer, we assume that: gz is defined by 2 with b > 0 + 1, 0 α <, N, 0 λ 1,β 0 and z U. Theorem 1 A function fz S,λ f, g; α, β if + λ [1 + β α + β] a b α. 20 =+1 Proof. It suffices to show that 1 λ + λ β zf z + λ g z2 f g z 1 λ f gz + λzf z g 21

40 M. K. Aouf, A. O. Mostafa and A. A. Hussain Re { 1 λ + λ zf } g z + λ z2 f g z 1 λ f gz + λzf α. 22 g z We have Re 1 λ + λ β zf g z + λ z2 f g z 1 λ f gz + λzf g z { 1 λ + λ zf } g z + λ z2 f g z 1 λ f gz + λzf g z 1 λ + λ 1 + β zf g z + λ z2 f g z 1 λ f gz + λzf g z 1 + β =+1 1 =+1 +λ +λ a b 23 24 25 a b. 26 The last exression is bounded above by α since 20 holds. Hence the roof of Theorem 1 is comleted. Theorem 2. A function fz T S,λ f, g; α, β if and only if =+1 + λ [1 + β α + β] a b α. 27 Proof. In view of Theorem 1, we need only to rove the necessity. fz T S,λ f, g; α, β and z is real, then +λ =+1 +λ =+1 1 a b z α β a b z +λ =+1 1 +λ =+1 Letting z 1 along the real axis, we obtain the desired inequality =1+ + λ and hence the roof of Theorem 2 is comleted. If a b z. a b z 28 [1 + β α + β] a b α 29

Certain Subclass of Starlie and Convex Uniformly Functions 41 Corollary 1 a If a function fz T S,λ f, g; α, β. Then +λ The result is shar for the function fz = z +λ 3 Distortion theorems α + 1. 30 [1 + β α + β] b α z + 1. 31 [1 + β α + β] b Theorem 3 If fz T S,λ f, g; α, β, then for z = r < 1 fz r +λ α r +1 + 1 + β αb +1 32 and fz r + +λ α r +1, 33 + 1 + β αb +1 rovided that b b +1 +1.The equalities in 32 and 33 are attained for the function fz given by fz = z +λ Proof. Using Theorem 2, we have + λ + 1 + β α b +1 that is, that =+1 + λ =+1 From 4 and 37, we have a α z +1. 34 + 1 + β αb +1 =+1 a 35 [1 + β α + β] b a α 36 +λ fz r r 1+ α. 37 + 1 + β αb +1 =+1 a 38

42 M. K. Aouf, A. O. Mostafa and A. A. Hussain and r r + +λ α r +1 + 1 + β αb +1 fz r + r 1+ +λ This comletes the roof of Theorem 3. =+1 39 a 40 α r +1. 41 + 1 + β αb +1 Theorem 4 Let fz T S,λ f, g; α, β, then for z = r < 1 f z r 1 and f z r 1 + +λ +λ + 1 α r 42 + 1 + β αb +1 + 1 α r, 43 + 1 + β αb +1 rovided that b b +1 + 1.The result is shar for the function fz given by 34. Proof. From Theorem 2, we have =+1 a +λ + 1 α. 44 + 1 + β αb +1 The remaining art of the roof is similar to the roof of Theorem 3, then we omit the details. Now, differentiating both sides of 4 m-times, we have f m z =! m! z m =+1! m! a z m +1; N; m <, m N 0. 45 Theorem 5 If a function fz T S,λ f, g; α, β. Then for z = r < 1, we have f m z! + 1! α r r m m! + 1 m! + 1 + β αb +1 +λ 46

Certain Subclass of Starlie and Convex Uniformly Functions 43 and f m z! m! + +λ + 1! α r r m. + 1 m! + 1 + β αb +1 The result is shar for the function fz given by 34. Proof. Using 27, we have =+1! m! a From 45 and 48, we have and +λ f m z! m! r m r +1 m! m! +λ 47 + 1! α. 48 + 1 m! + 1 + β αb +1 =+1! m! a 49 + 1! α r r m 50 + 1 m! + 1 + β αb +1 f m z! m! r m + r +1 m! m! + +λ This comletes the roof of Theorem 5. =+1! m! a 51 + 1! α r r m 52 + 1 m! + 1 + β αb +1 Remar 1 Putting m = 0 and m = 1, in Theorem 5 we obtain Theorem 3 and Theorem 4, resectively. 4 Convex linear combinations Theorem 6 Let η µ 0 for µ = 1, 2,..., n and n η µ 1. If the functions f µ z defined by f µ z = z a,µ z a,µ 0; µ = 1, 2,..., n 53 =+1 µ=1

44 M. K. Aouf, A. O. Mostafa and A. A. Hussain are in the class T S,λ f, g; α, β for every µ = 1, 2,..., n then the function fz defined by n fz = z η µ a,µ z 54 is in the class T S,λ f, g; α, β. =+1 Proof. Since f µ z T S,λ f, g; α, β, it follows from Theorem 2 that =+1 =+1 + λ µ=1 [1 + β α + β] b a,µ α, 55 for every µ = 1, 2,..., n. Hence n + λ [1 + β α + β] η µ a,µ b 56 = n η µ µ=1 =+1 + λ α µ=1 [1 + β α + β] a,µ b 57 n η µ α. 58 µ=1 By Theorem 2, it follows that fz T S,λ f, g; α, β and hence the roof of Theorem 6 is comleted. The class T S,λ f, g; α, β is closed under convex linear combi- Corollary 2 nations. Theorem 7 Let f z = z and f z = z +λ Then fz T S,λ f, g; α, β if and only if where γ 0 and γ = 1. = α z + 1. 59 [1 + β α + β] b f z = =+1 γ f z, 60

Certain Subclass of Starlie and Convex Uniformly Functions 45 Proof. Assume that f z = γ f z 61 = = z =+1 +λ α γ z. 62 [1 + β α + β] b Then it follows that +λ [1 + β α + β] b α =+1 = =+1 +λ α [1 + β α + β] b 63 γ = 1 γ 1. 64 γ So, by Theorem 2, fz T S,λ f, g; α, β. Conversely, assume that the function fz belongs to the class T S,λ f, g; α, β. Then a +λ α + 1. 65 [1 + β α + β] b Setting and γ = +λ [1 + β α + β] a b α + 1 66 γ = 1 γ, 67 =+1 we see that fz can be exressed in the form 60. This comletes the roof of Theorem 7. Corollary 3 The extreme oints of the class T S,λ f, g; α, β are the functions f z = z and f z = z +λ α z + 1. 68 [1 + β α + β] b

46 M. K. Aouf, A. O. Mostafa and A. A. Hussain 5 Radii of close-to-convexity, starlieness and convexity Theorem 8 Let the function fz T S,λ f, g; α, β. Then Corollary 4 i fz is -valent close-to-convex of order δ 0 δ < in z < r 1, where r 1 = inf +1 +λ δ [1 + β α + β] b α 1. 69 ii fz is -valent starlie of order δ 0 δ < in z < r 2, where r 2 = inf +1 +λ δ [1 + β α + β] b δ α The result is shar, the extremal function being given by 31. 1. 70 Proof. i We must show that f z z 1 δ for z < r 1, 71 where r 1 is given by 69. Indeed we find from 4 that Thus if f z z 1 =+1 f z z 1 =+1 a z. 72 δ, 73 a z 1. 74 δ But, by Theorem 2, 74 will be true if +λ [1 + β α + β] b z, 75 δ α

Certain Subclass of Starlie and Convex Uniformly Functions 47 that is, if z +λ δ [1 + β α + β] b α 1 + 1. 76 the roof of i is comleted. ii It is sufficient to show that zf z fz δfor z < r 2, 77 where r 2 is given by 70. Indeed we find, again from 4 that a zf z fz z =+1 1. 78 a z Thus if =+1 zf z fz δ 79 =+1 But, by Theorem 2, 80 will be true if that is, if z δ δ z +λ the roof of ii is comleted. δ δ z a 1. 80 +λ [1 + β α + β] b α δ [1 + β α + β] b δ α 1 81 + 1. 82 Corollary 5 Let fz T S,λ f, g; α, β. Then fz is -valent convex of order δ 0 δ < in z < r 3, where 1 +λ δ [1 + β α + β] b r 3 = inf. 83 +1 δ α The result is shar, with the extremal function fz given by 31.

48 M. K. Aouf, A. O. Mostafa and A. A. Hussain 6 Classes of reserving integral oerators In this section, we discuss some classes reserving integral oerators. Consider the following oerators i F z defined by where ii Gz defined by F z = J c, fz = c + z c = z d = = z =+1 z 0 t c 1 f tdt 84 d z f S; c >, 85 c + a a + 1. 86 c + z Gz = z 1 =+1 iiithe Komatu oerator [16] defined by Hz = P d c,fz = = z =+1 0 ft t dt 87 1 + 1 a z + 1. 88 c + d Γdz c z 0 d c + a z c + t c 1 log z t d 1 f tdt d > 0; c > ; N; z U, 89 and iv Iz, which generalizes Jung-Kim-Srivastava integral oerator see [13] defined by Iz = Q d Γd + c + 1 c,fz = Γc + Γd z c = z + =+1 z Γc + Γd + c + Γc + Γ d + c + a z 0 t c 1 1 t d 1 f tdt z

Certain Subclass of Starlie and Convex Uniformly Functions 49 d > 0; c > ; z U, 90 In view of Theorem 2, we see that z d z is in the class T S,λ f, g; α, β =+1 as long as 0 d a for all. In articular, we have Theorem 9 Let fz T S f, g; α, β, λ and c be a real number such that c >. Then the function F z defined by 84 also belongs to the class T S,λ f, g; α, β. Proof. From 27 and 86, we have the function F z is in the class T S,λ f, g; α, β. Theorem 10 Let F z = z =+1 a z a 0 be in the class T S,λ f, g; α, β, and let c be a real number such that c >.Then the function fz given by 84 is -valent in z < R 1, where R 1 = inf +1 The result is shar. { c + + λ [1 + β α + β] b Proof. From 84, we have c + α } 1. 91 fz = z1 c [z c F z] c > 92 c + c + = z a z. 93 c + =+1 In order to obtain the required result, it suffices to show that f z z 1 < whenever z < R 1, 94 where R 1 is given by 91. Now f z z 1 Thus f z z 1 < if =+1 =+1 c + c + a z. 95 c + c + a z < 1. 96

50 M. K. Aouf, A. O. Mostafa and A. A. Hussain But Theorem 2 confirms that +λ [1 + β α + β] a b 1. 97 α =+1 Hence 96 will be satisfied if that is, if c + c + z < [ + λ ] [1 + β α + β] b, 98 α { } 1 c + [ + λ ] [1 + β α + β] b z < + 1. 99 c + α Therefore, the function fz given by 84 is -valent in z < R 1. Sharness of the result follows if we tae fz = z +λ αc + z + 1 100 [1 + β α + β] b c + and hence the roof of Theorem 10 is comleted. Theorem 11 If fz T S,λ f, g; α, β, then Hz T S,λ f, g; α, β. if Proof. The function Hz defined by 6.5 is in the class T S,λ f, g; α, β =+1 [+λ ] [1 + β α + β] b α c+ d c+ a 1. 101 Since c+ c+ 1 for + 1, then =+1 =+1 [+λ ] [+λ ] Therefore Hz T S,λ f, g; α, β. [1 + β α + β] b c+ d c+ α [1 + β α + β] b α a 102 a 1. 103

Certain Subclass of Starlie and Convex Uniformly Functions 51 Theorem 12 Let d > 0, c > and fz T S,λ f, g; α, β. Then function Hz defined by 89 is -valent in z < R 2, where R 2 = inf +1 { c + d + λ [1 + β α + β] b c + d α The result is shar for function fz given by fz = z +λ } 1. 104 αc + d [1 + β α + β] b c + d z + 1. 105 Proof. In order to rove the assertion. it is enough to show that H z z 1 < in z < R 2 106 Now H z z 1 d c + a z c + d c + a z. c + =+1 =+1 The last inequality is bounded a above by if H z z 1 d c + a z 1. 107 c + =+1 Since fz T S,λ f, g; α, β, then in view of Theorem 2, we have [+λ ] [1 + β α + β] b a 1. 108 α =+1 Thus, 107 holds if or d c + z [ + λ ] [1 + β α + β] b c + α, 109 { } c + d 1 [ + λ ] [1 + β α + β] b z + 1. c + d α 110

52 M. K. Aouf, A. O. Mostafa and A. A. Hussain Therefore, the function Hz given by 89 is -valent in z < R 2. This comletes the rove of Theorem 12. Following similar stes as in the roofs of Theorems 11 and 12, we can rove the following theorems concerning the integral oerator Iz defined by 90. Theorem 13 If fz T S,λ f, g; α, β, then Iz T S,λ f, g; α, β,where Iz is given by 90. Theorem 14 Let d > 0, c > and fz T S,λ f, g; α, β. Then function Iz defined by 90 is -valent in z < R 3, where R 3 = inf +1 { Γc + Γ d + c + + λ [1 + β α + β] b Γc + Γd + c + α The result is shar for function fz given by fz = z +λ } 1. 111 αγc + Γd + c + z +1. [1 + β α + β]b Γc + Γ d + c + 7 Modified Hadamard roducts 112 Let the functions f µ z be defined for µ = 1, 2,...m by 53. The modified Hadamard roducts of f 1 z and f 2 z is defined by f 1 f 2 z = z =+1 a,1 a,2 z. 113 Theorem 15 Let f µ z T S,λ f, g; α, βµ = 1, 2, where f µ zµ = 1, 2are in the form 53. Then f 1 f 2 z T S,λ f, g; δ, β, where δ = The result is shar for f µ z = z +λ +λ α 2 1 + β. 114 + 1 α + β 2 b +1 α 2 α z +1 µ = 1, 2. 115 + 1 α + βb +1

Certain Subclass of Starlie and Convex Uniformly Functions 53 Proof. Emloying the technique used earlier by Schild and Silverman [21], we need to find the largest real arameter δ such that +λ [1 + β δ + β]b a,1 a,2 1. 116 δ =+1 Since f µ T S,λ f, g; α, βµ = 1, 2, we have +λ [1 + β α + β]b a,1 1 117 α =+1 and =+1 +λ [1 + β α + β]b α a,2 1. 118 By the Cauchy-Schwarz inequality we have +λ [1 + β α + β]b a,1 a,2 1. 119 α =+1 thus it sufficient to show that +λ [1 + β δ + β]b a,1 a,2 120 δ +λ [1 + β α + β]b a,1 a,2 121 α or, equivalently, that a,1 a,2 δ[1 + β α + β] α[1 + β δ + β]. 122 Hence, in the light of the inequality 119, it is sufficient to rove that +λ α δ[1 + β α + β] [1 + β α + β]b α[1 + β δ + β]. 123 It follows from 123 that δ 1 + β α 2 [1 + β α + β] 2 [+λ ] b α. 124 2

54 M. K. Aouf, A. O. Mostafa and A. A. Hussain Let M = +λ 1 + β α 2, 125 [1 + β α + β] 2 b α 2 then M is increasing function of + 1. Therefore, we conclude that δ M + 1 = +λ and hence the roof of Theorem 15 is comleted. 1 + β α 2 126 + 1 + β α 2 b +1 α 2 Theorem 16 Let f 1 z T S,λ f, g; α, β and f 2 z T S,λ f, g; γ, β, where f µ zµ = 1, 2 are in the form 53. Then f 1 f 2 z T S,λ f, g; ξ, β,where 1 + β α γ ξ. 127 +λ + 1 + β α + 1 + β γb +1 α γ The result is shar for f 1 z = z +λ α z +1 + 1 + β αb +1 128 and f 2 z = z +λ γ z +1. 129 + 1 + β γb +1 Proof. We need to find the largest real arameter ξ such that +λ [1 + β ξ + β]b a,1 a,2 1. 130 ξ =+1 Since f 1 z T S,λ f, g; α, β, we readily see that +λ [1 + β α + β]b a,1 1 131 α =+1 and f 2 z T S,λ f, g; γ, β, we have +λ [1 + β γ + β]b a,2 1. 132 γ =2

Certain Subclass of Starlie and Convex Uniformly Functions 55 By the Cauchy-Schwarz inequality we have +λ [1 + β α + β] 1 2 [1 + β γ + β] 1 2 b a,1 a,2 1 α γ =2 133 thus it is sufficient to show that +λ [1 + β ξ + β]b a,1 a,2 134 ξ +λ [1 + β α + β] 1 2 [1 + β γ + β] 1 2 b a,1 a,2 α γ 135 or, equivalently, that a,1 a,2 ξ[1 + β α + β] 1 2 [1 + β γ + β] 1 2 α γ[1 + β ξ + β]. 136 Hence, in the light of the inequality 133, it is sufficient to rove that α γ 137 +λ [1 + β α + β] 1 2 [1 + β γ + β] 1 2 b ξ [1 + β α + β] 1 2 [1 + β γ + β] 1 2. 138 α γ [1 + β ξ + β] It follows from 137 that ξ Let A = +λ +λ 1 + β α γ. [1 + β α + β][1 + β γ + β]b α γ 139 1 + β α γ, [1 + β α + β][1 + β γ + β]b α γ 140 then A is increasing function of + 1. Therefore, we conclude that 1 + β α γ ξ A+1 = +λ + 1 + β α + 1 + β γb +1 α γ 141 and hence the roof of Theorem 16 is comleted.

56 M. K. Aouf, A. O. Mostafa and A. A. Hussain Theorem 17 Let f µ z T S,λ f, g ; α, βµ = 1, 2, 3, where f µ zµ = 1, 2, 3 are in the form 53. Then f 1 f 2 f 3 z T S,λ f, g ; τ, β, where τ +λ 1 + β α 3 2. 142 + 1 + β α3 b 2 +1 α 3 The result is best ossible for functions f µ zµ = 1, 2, 3 given by 115. Proof. From Theorem 15, we have f 1 f 2 z T S,λ f, g ; δ, β, where δ is given by 114. Now, using Thoerem 16, we get f 1 f 2 f 3 z T S,λ f, g ; τ, β, where τ 1+λ 1 + β α δ. [1 + β α + β][1 + β δ + β]b α δ Now defining the function B by 143 B = +λ 1 + β α δ. [1 + β α + β][1 + β δ + β]b α δ 144 We see that B is increasing function of +1. Therefore, we conclude that 1 + β α δ τ B+1 =, +λ + 1 + β α + 1 + β δb +1 α δ 145 substituting from 7.2, we have τ = +λ and hence the roof of Theorem 17 is comleted. 1 + β α 3 2 146 + 1 + β α3 b 2 +1 α 3 Theorem 18 Let f µ z T S,λ f, g; α, βµ = 1, 2, where f µ zµ = 1, 2are in the form 53. Then h z = z a 2,1 + a 2,2z 147 =+1

Certain Subclass of Starlie and Convex Uniformly Functions 57 belongs to the class T S,λ f, g; ϕ, β, where ϕ +λ 21 + β α 2. 148 + 1 + β α 2 b +1 2 α 2 The result is shar for functions f µ zµ = 1, 2 defined by 115. and Proof. By using Theorem 16, we obtain +λ [1 + β α + β]b =+1 =+1 =2 +λ =2 +λ +λ α [1 + β α + β]b α a,1 [1 + β α + β]b α [1 + β α + β]b α It follows form 149 and 151 that 1 +λ [1 + β α + β] b 2 α =2 Therefore, we need to find the largest ϕ such that +λ [1 + β ϕ + β]b 1 2 +λ 2 a,2 2 2 2 a 2,1 149 1, 150 a 2,2 151 ϕ [1 + β α + β] b α 2 1. 152 a 2,1 + a 2,2 1. 153 2 154, 155 that is ϕ +λ 21 + β α 2 156 [1 + β α + β]b 2 α 2

58 M. K. Aouf, A. O. Mostafa and A. A. Hussain Let H = +λ 21 + β α 2, 157 [1 + β α + β] 2 b 2 α 2 then H is increasing function of + 1. Therefore, we conclude that ϕ H + 1 = +λ and hence the roof of Theorem 18 is comleted. 21 + β α 2, 158 + 1 + β α 2 b +1 2 α 2 Remar 2 i Putting gz = z + C, n1 γ n z, where C, n given =1 by 16 by relacing by, λ = 0 and λ = 1, resectively, in Theorems 16, 17 and??, resectively, we obtain modified Hadamard roducts for the classes Sα, β, λ and C α, β, λ, resectively, defined in the introduction. ii Putting λ = 0 and b = Ω, where Ω is given by11 and 0 α <, β 0, α i Ci = 1, 2,...l, β j C\{ 1, 2,...}j = 1, 2,...min Theorems 16, 17 and?? resectively we modified the results obtained by Marouf [16, Theorems 4, 3 and 5, resectively]. iii Putting λ = 0 and b = φ n.λ.δ,, where φ is given by 13 and n N 0, λ 0, δ 0, α <, β 0in Theorems 16, 17 and??, resectively, we obtain the results obtained by Salim et al. [20, Theorems 3, 2 and 4, resectively]. iiv Putting λ = 0 and b = Φ,where Φ is given by 17 and < µ < 1, < γ < 1, η R +, N, α <, β 0, a R, c R\Z, z U,in Theorems 16, 17 and??, resectively, we obtain the results obtained by Khairnar and More[15, Thoerems 4.2, 4.1 and 5.1, resectively, with n = 1]. Remar 3 Secializing the arameters, λ, α, β and function gzin our results,we obtain new resultes associated to the subclasses T S,λ n, α, β and T S,n λ,q,s n, α, β defined in the introduction. 8 Oen roblem The authors suggest to obtain the same roerties of the class consisting of functions fz, gz T and satisfying the subordinating condition: 1 γ + γ zf g z + γ z2 f g z 1 γ f gz + γ zf g z 159

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