SOME CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING CERTAIN LINEAR OPERATOR

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1 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 56 SOME CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING CERTAIN LINEAR OPERATOR Abdul Rahman S Juma 1, Hazha Zirar Hussain 2 Deartment of Mathematics, Alanbar University, Ramadi-Iraq 1, Deartment of Mathematics, Salahaddin University,Erbil-Iraq 2, Abstract Making use of a linear oerator, which is defined here by means of the Hadamard roduct (or convolution), we introduce two novel subclasses D a,c,λ [, α, A, B] and [α, β] of meromorhically multivalent functions In this aer, we obtain coefficient estimates, distortion theorems, radii of starlikeness and convexity and closure theorems for the class D a,c,λ [, α, A, B] Several interesting results involving the Hadamard roduct of functions belonging to the class D a,c,λ [, α, A, B], [α, β] and [α, β] are also derived Also integral transforms of functions in the classes [α, β] and [α, β] are studied Key Words : Linear oerator, Meromorhic, Positive coefficients, Hadamard roduct AMS Subject Classification : Secondary 30C45 1 Introduction Let denote the class of functions of the form: f(z) = 1 z a n z n, (a n 0; N = {1, 2, ), (1) which are analytic and -valent in the unctured unit disk U = {z C : 0 < z < 1 = U {0; where U = {z C : z < 1 For functions f(z) given by (1) and g(z) given by g(z) = 1 z b n z n, (b n 0), 1 dr juma@hotmailcom 2 hazhazirar@yahoocom

2 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 57 we define the Hadamard roduct (or convolution) of two functions, f(z) and g(z) given by (f g)(z) = 1 z a n b n z n = (g f)(z) In terms of the Pochhammer symbol (θ) n given by (θ) n = Γ(θ n) Γ(θ) { = we define the function ϕ(a, c, ; z) by 1 (n = 0), θ(θ 1)(θ 2)(θ n 1) (n N), ϕ(a, c, ; z) = 1 z (z U ; a R; c R Z 0 ; Z 0 = {0, 1, 2, ) z n (2) Corresonding to the function ϕ(a, c, ; z), Liu [8] and Liu and Srivastava [9] have introduced a linear oerator l (a, c) which is defined by means of the following Hadamard roduct (or convolution): l (a, c) = ϕ(a, c, ; z) f(z) (3) Just as in [8] and [9], it is easily verified from the definitions (2) and (3) that z(l (a, c))f(z)) = al (a 1, c)f(z) (a )l (a, c)f(z) We also note, for any integer m > and for f(z), that l (n, 1)f(z) = D m 1 f(z) = 1 f(z), z (1 z) m where D m 1 f(z) is the differential oerator studied by (among others) Uralegaddi and Somanatha [17] and Aouf [3] Further M K Aouf et al [4] considered the generalized oerators as follows: Let F (z) = (1 λ)l (a, c)f(z) λ z(l (a, c)f(z))

3 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 58 (f ; N; 0 λ < 1 2 ), so that, obviously, F (z) = 1 2λ z ( N; 0 λ < 1 2 ), [1 λ( n ] a n z n (4) since f(z) is given by (1) From (4), it is easily verified that zf (z) = af,a1,c,λ (z) (a )F (z) We say that a function f(z) is in the class D a,c,λ[, α, A, B] if it satisfies the following inequality: B zf (z) F (z) zf (z) F (z) [B (A B)( α)] < 1 (z U ), (5) where the arameters A, B, α, and λ are constrained as follows: 1 A < B 1, A B 0, 0 α <, N and 0 λ < 1 2 The class D a,a,0 [, α, A, B] = Q [, α, A, B] was studied by Aouf [2] and Srivastava et al[15] We observe also that 1 D a,c,λ [, α, β, β] = [α, β] = {f(z) :, N, 0 < β 1, 0 α <, z U zf (z) F (z) zf (z) F (z) 2α < β, 0 α < 2 D a,a,0 [, α, β, β] = [α, β], the class of meromorhic -valent starlike functions of order α and tye β, = {f(z) zf (z) f(z) : f (z) 2α < β, 0 α <, N, 0 < β 1, z U f(z)

4 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 59 Meromorhically multivalent functions have been extensively studied by (for examle) Mogra [10, 11], Uralegaddi and Ganigi [16], Aouf [1, 2], Srivastava et al [15], Owa et al [12], Joshi and Srivastava [7], Liu [8], Liu and Srivastava [9], Aouf et al [5], Raina and Srivastava [13] and Yang [18] In this aer we investigate various imortant roerties and characteristics of the class D a,c,λ [, α, A, B], we obtain coefficient estimates, distortion theorems, radii of starlikeness and convexity and closure theorems Several interesting results involving the Hadamard roduct of functions belonging to the classes D a,c,λ [, α, A, B], [α, β] and [α, β] are also derived Also integral transforms for functions in the classes [α, β] and [α, β] are studied 2 Coefficient Estimates Theorem 21 : A function f(z) defined by (1) is said to be in the class D a,c,λ [, α, A, B] if and only if [1λ( n )] [(1B)n22αB(BA)( α)]a n (1 2λ)(B A)( α), (6) where 1 A < B 1, A B 0, 0 α <, N and 0 λ < 1 2 roof : We assume that the inequality (6) holds true Then, if we let z U, we find from (1) and (6) that B zf (z) F (z) zf (z) F (z) [B (A B)( α)] [1 λ( n )] (2 n)a n (1 2λ)(B A)( α) [1 λ( n )] [B(n 2α) (B A)( α)]a n 1(z U = {z C : z = 1 Hence, by the maximum modulus theorem we have f(z) D a,c,λ [, α, A, B]

5 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 60 Conversely, let f(z) D a,c,λ [, α, A, B] be given by (1) Then, from (1) and (6), we have zf (z) F (z) [B (A B)( α)] = B zf (z) F (z) [1 λ( n )] (2 n)a n z 2n (1 2λ)(B A)( α) [1 λ( n ] < 1 for z U [B(n 2α) (B A)( α)]a n z 2n Since R(z) z (z C), we have { R [1 λ( n)] (a) n1 (2 n)a n z 2n (1 2λ)(B A)( α) [1 λ( n)] < 1 [B(n 2α) (B A)( α)]a n z 2n Choose values of z on the real axis so that zf (z) F (z) in (7) and letting z 1 through real values, we obtain (7) Hence the roof is comlete (7) is real Uon clearing the denominator Corollary 21 Let the function f(z) defined by (1) be in the class D a,c,λ [, α, A, B] Then a n (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)], (n 0) The result is shar for the function: f(z) = 1 z (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] z n (8) Putting A = β and B = β(0 < β 1) in Theorem 21, we obtain: Corollary 22 A function f(z) defined by (1) is in the class [α, β] if and only if [1 λ( n ] [(1 B)n 2( αβ)]a n (1 2λ)2β( α) Putting λ = 0 and a = c in Corollary 22, we obtain: Corollary 23 A function f(z) defined by (1) is in the class [α, β] if and only if [(1 B)n 2( αβ)]a n 2β( α)

6 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 61 Putting A = 1, B = 1, a = c and λ = 0 in Theorem 21, we obtain: Corollary 24 A function f(z) defined by (1) is in the class [α] if and only if (n α)a n ( α) 3 Distortion Theorem Theorem 31 : If the function f(z) defined by (1) is in the class D a,c,λ [, α, A, B], then for 0 < z = r < 1, we have 1 c(1 2λ)(B A)( α) r a[2 2αB (B A)( α)] r f(z) 1 c(1 2λ)(B A)( α) r a[2 2αB (B A)( α)] r The result is shar roof : In view of Theorem 21, we have a c [22αB(BA)( α)] a n that is, that a n Then, for o < z = 1 < 1, [1λ( n )] [(1B)n22αB(BA)( α)]a n (1 2λ)(B A)( α), (1 2λ)(B A)( α) [2 2αB (B A)( α)] c a f(z) = 1 z a n z n, f(z) 1 r a n r n, 1 r r a n 1 (1 2λ)(B A)( α) r [2 2αB (B A)( α)] c a r

7 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 62 and f(z) = 1 z a n z n, f(z) 1 r a n r n, 1 r r a n 1 (1 2λ)(B A)( α) r [2 2αB (B A)( α)] c a r The bounds for f(z) are shar and are attained for the function at z = r, z = re iπ 2 Hence the roof is comlete f(z) = 1 (1 2λ)(B A)( α) z [2 2αB (B A)( α)] c a z Next we roof the following growth and distortion roerties for the class D a,c,λ [, α, A, B] Theorem 32 : If the function f(z) defined by (1) is in the class D a,c,λ [, α, A, B] Then ( m 1)! r (m) c!(1 2λ)(B A)( α) ( 1)! a( m)![2 2αB (B A)( α)] rn m f (m) (z) ( m 1)! r (m) c!(1 2λ)(B A)( α) ( 1)! a( m)![2 2αB (B A)( α)] rn m (0 < z = r < 1; a > c > 0; m N 0 = N {0; N; > m) The result is shar for the function f(z) given by at z = r, z = re iπ 2 f(z) = 1 c(1 2λ)(B A)( α) z a[2 2αB (B A)( α)] z roof : In view of Theorem 21, we have a!c [2 2αB (B A)( α)] ( n)!a n

8 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 63 which yields [1 λ( n )] [(1 B)n 2 2αB (B A)( α)]a n ( n)!a n (1 2λ)(B A)( α), c!(1 2λ)(B A)( α) ( N) (9) a[2 2αB (B A)( α)] Now, by differentiating both sides of (1) m times with resect to z, we have f (m) m ( m 1)! (z) = ( 1) z (m) ( n)! ( 1)! ( n m)! a nz n m ( N, m N 0 ; > m), (10) and Theorem 32 follows easily from (9) and (10) Hence the roof is comlete Next we determine the radii of meromorhically -valent starlikeness of order δ(0 δ < ) and meromorhically -valent convexity of order δ(0 δ < ) for functions in the class D a,c,λ [, α, A, B] 4 Radii of Sarlikness and Convexity Theorem 41 : Let the function f(z) defined by (1) be in the class D a,c,λ [, α, A, B] Then 1 f(z) is meromorhically -valent starlike of order δ(0 δ < ) in the disk z < r 1, that is, R{ zf (z) f(z) > δ ( z < r 1; 0 δ <, n N), where r 1 (, α, A, B, a, c) = [ ( δ)[1λ( n )][(1B)n22αB(BA)( α)] (nδ)(1 2λ)(B A)( α) ] 1 (a) 2n n1 (n 0) 2 f(z) is meromorhically -valent convex of order δ(0 δ < ) in the disk z < r 1, that is, R{ (1 zf (z) f (z)) > δ ( z < r 2; 0 δ <, n N),

9 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 64 where r 2 (, α, A, B, a, c) = [ ( δ)[1λ( n ] )][(1B)n22αB(BA)( α)] 1 (a) 2n n1 (n)(nδ)(1 2λ)(B A)( α) (n 0) Each of these results is shar for the function f(z) given by (8) roof : 1 It is sufficient to show that Note that zf (z) f(z) zf (z) f(z) 2δ Thus, we have the desired inequality if zf (z) f(z) zf (z) f(z) But Theorem 21 ensures that zf (z) f(z) zf (z) f(z) 2δ 1 (2 n)a n z n 2( δ) (n 2δ)a n z 2n 1 (0 δ <, n N), (11) 2δ (n δ) ( a n z 2n 1 ( δ) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n 1 (12) (1 2λ)(B A)( α) In view of (12), it follows that (11) will be true if or if ( n δ ) r 2n [1 λ( n)] [(1 B)n 2 2αB (B A)( α)], δ (1 2λ)(B A)( α) r [ ( δ)[1 λ( n (n δ)(1 2λ)(B A)( α) )][(1 B)n 2 2αB (B A)( α)] ] 1 2n (n 0) (13) Setting r = r 1 (, α, A, B, λ, a, c) in (13), the result follows

10 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No: In order to rove the second assertion of Theorem 41, it sufficient to show that 1 zf (z) f (z) 1 (0 δ <, n N) 1 1f (z) 2δ f (z) Note that 1 zf (z) f (z) 1 f (z) 2δ f (z) (n )(n 2)a n z 2n 2( δ) ( n)(n 2δ)a n z 2n Thus we have the desired inequality 1 zf (z) f (z) 1 f (z) 2δ 1 (0 δ <, n N), f (z) By Theorem 21, (14) will be true if ( n)( n δ) r 2n [1 λ( n ( δ) or if r [ ( δ)[1 λ( n )] ( n)( n δ) a n r 2n 1 (14) ( δ) )][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) [(1 B)n 2 2αB (B A)( α)] ( n)( n δ)(1 2λ)(B A)( α) ] 1 2n (n 0) Setting r = r 2 (, α, A, B, λ, a, c) in (15), the result follows, and the roof of Theorem 41 comleted by merely verifying that each assertion is shar for the function f(z) given by (8) 5 Closure Theorems Theorem 51 : Let and f 1 (z) = 1 z, f n (z) = 1 z (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] z n, (n 0) (15)

11 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 66 Then f(z) is in the class D a,c,λ [, α, A, B] if and only if it can be exressed in the form f(z) = where µ n 0 and n= 1 µ n = 1 n= 1 µ n f n (z), roof : First suose that f(z) can be exressed of the form Then = 1 z f(z) = n= 1 µ n f n (z) µ n (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] z n µ n (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) = µ n = 1 µ 1 1, which shows that f(z) D a,c,λ [, α, A, B] Conversely, suose that f D a,c,λ [, α, A, B] Then a n and setting and (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (n 0), µ n = [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n (n 0), (1 2λ)(B A)( α) µ 1 = 1 it follows that f(z) = n= 1 µ nf n (z) Hence the roof is comlete λ n,

12 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 67 Theorem 52 : The class f D a,c,λ [, α, A, B] is closed under convex linear combinations roof : Let each of the functions f j (z) = 1 z a n,j z n, (j = 1, 2) (16) be in the class D a,c,λ [, α, A, B] It is sufficient to show that the function h(z) defined by h(z) = (1 t)f 1 (z) tf 2 (z), (0 t 1) is also in the class D a,c,λ [, α, A, B] Since h(z) = 1 z [(1 t)a n,1 ta n,2 ]z n (0 t 1), with the aid of Theorem 21, we have [1 λ( n )] [(1 B)n 2 2αB (B A)( α)][(1 t)a n,1 ta n,2 ] = (1 t) t [1 λ( n )] [(1 B)n 2 2αB (B A)( α)]a n,1 [1 λ( n )] [(1 B)n 2 2αB (B A)( α)]a n,2 (1 t)(1 2λ)(B A)( α) t(1 2λ)(B A)( α) which shows that h(z) D a,c,λ [, α, A, B] This comletes the roof of the theorem 6 Convolution Proerties = (1 2λ)(B A)( α), For functions f j (z)(j = 1, 2) defined by (16) belonging to the class, we denote by (f 1 f 2 )(z) the convolution (or Hadamard roduct) of the functions f 1 (z) and f 2 (z); that is (f 1 f 2 )(z) = 1 z a n,1 a n,2 z n

13 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 68 Theorem 61 : Let the functions f 1 (z) defined by (16) be in the class D a,c,λ [, α, A, B], and the function f 2 (z) defined by (16) be in the class D a,c, [, γ, A, B] Then (f 1 f 2 )(z) D a,c,λ [, τ, A, B], where { τ 1 2c(1 2λ)(1B)(B A)( α)( γ) a[22αb(ba)( α)][22γb(ba)( γ)](1 2λ)(B A) 2 ( α)( γ) The result is shar for the functions f j (z)(j = 1, 2) given by f 1 (z) = 1 c(1 2λ)(B A)( α) z a[2 2αB (B A)( α)] z, ( N) f 2 (z) = 1 c(1 2λ)(B A)( γ) z a[2 2γB (B A)( γ)] z, ( N) roof : Emloying the technique used earlier by Schild and Silverman [13], we need to find the largest τ such that [1 λ( n )][(1 B)n 2 2τB (B A)( τ)] a n,1 a n,2 1, (1 2λ)(B A)( τ) for f 1 (z) D a,c,λ [, α, A, B], and f 2 (z) D a,c,λ [, γ, A, B] Since f 1 (z) D a,c,λ [, α, A, B], and f 2 (z) D a,c,λ [, γ, A, B], we readily see that [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n,1 1, (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2γB (B A)( γ)] a n,2 1 (1 2λ)(B A)( γ) Therefore, by the Cauchy- Schwarz inequality, we obtain 1(17) [1λ( n )] [(1B)n22αB(BA)( α)][(1b)n22γb(ba)( γ)] (1 2λ)(B A)) ( α)( γ) This imlies that we need only to show that an,1 a n,2 [(1 B)n 2 2τB (B A)( τ)] a n,1 a n,2 ( τ) [(1B)n22αB(BA)( α)][(1b)n22γb(ba)( γ)] a ( α)( γ) n,1 a n,2 (n 0), or equivalently, that

14 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 69 an,1 a n,2 ( τ) [(1B)n22αB(BA)( α)][(1b)n22γb(ba)( γ)] ( α)( γ)[(1b)n2τb(ba)( τ)] (n 0) Hence, by the inequality (17), it is sufficient to rove that (1 2λ)(B A) ( α)( γ) [1λ( n )] [(1B)n22αB(BA)( α)][(1b)n22γb(ba)( γ)] ( τ) [(1 B)n 2 αb (B A)( α)][(1 B)n 2 2γB (B A)( γ)] ( α)( γ)[(1 B)n 2 2τB (B A)( τ)] (n 0) It follows from (18) that (18) τ 0) (n2)(1 2λ)(1B)(B A)( α)( γ) [1λ( n )][(1B)n22αB(BA)( α)][(1b)n22γb(ba)( γ)](1 2λ)(B A)2 ( α)( γ) (n Now, defining the function ϕ(n) by ϕ(n) = (n2)(1 2λ)(1B)(B A)( α)( γ) (c) [1λ( n n1 )][(1B)n22αB(BA)( α)][(1b)n22γb(ba)( γ)](1 2λ)(B A)2 ( α)( γ) We see that ϕ(n) is an increasing function of n(n 0) Therefore, we conclude that { τ ϕ(0) = 1 Hence the roof is comlete 2c(1 2λ)(1B)(B A)( α)( γ) a[22αb(ba)( α)][22γb(ba)( γ)](1 2λ)(B A) 2 ( α)( γ)] Putting A = β and B = β(0 < β 0) in Theorem 61, we obtain: Corollary 61 : Let the functions f 1 (z) defined by (16) be in the class [α, β], and the function f 2 (z) defined by (16) be in the class [γ, β] Then (f 1 f 2 )(z) [τ, β], where τ { 1 2c(1 2λ)β (1 B)( α)( γ) a( αβ)( γβ) (1 2λ)β 2 ( α)( γ) The result is shar for the functions f j (z)(j = 1, 2) given by f 1 (z) = 1 z f 2 (z) = 1 z Putting λ = 0 in Corollary 61, we obtain c(1 2λ)β( α) z, ( N) a( αβ) c(1 2λ)β( γ) z, a( γβ) ( N)

15 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 70 Corollary 62 : Let the functions f 1 (z) defined by (16) be in the class [α, β], and the function f 2 (z) defined by (16) be in the class [γ, β] Then (f 1 f 2 )(z) [τ, β], where { cβ(1 β)( α)( γ) τ 1 a( αβ)( γβ) β 2 ( α)( γ) The result is shar for the functions f j (z)(j = 1, 2) given by f 1 (z) = 1 cβ( α) z a( αβ) z, ( N) f 2 (z) = 1 cβ( γ) z a( γβ) z, ( N) result: Using an argument similar to those in the roof of Theorem 61, we obtain the following Theorem 62 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class D a,c,λ [, α, A, B] Then (f 1 f 2 )(z) D a,c,λ [, γ, A, B], where { 2c 2 (1 2λ)(1 B)(B A)( α) 2 γ 1 a 2 [2 2αB (B A)( α)] 2 (1 2λ)[(B A)( α)] 2 The result is shar for the functions f j (z)(j = 1, 2) given by f j (z) = 1 c(1 2λ)(1 B)(B A)( α) z a[2 2αB (B A)( α)] z, (j = 1, 2; N) (19) Putting A = β and B = β(0 < β 1) in Theorem 62, we obtain: Corollary 63 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class [α, β] Then (f 1 f 2 )(z) [γ, β], where { c(1 2λ)β(1 B)( α) 2 γ 1 a( αβ) 2 (1 2λ)β 2 ( α) 2 The result is shar for the functions f j (z)(j = 1, 2) given by f j (z) = 1 z c(1 2λ)β( α) z, (j = 1, 2; N) a( αβ)

16 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 71 Putting λ = 0 in Corollary 63, we obtain: Corollary 64 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class [α, β] Then (f 1 f 2 )(z) [γ, β], where { γ 1 cβ(1 β)( α) 2 a( αβ) 2 β 2 ( α) 2 The result is shar for the functions f j (z)(j = 1, 2) given by f j (z) = 1 cβ( α) z a( αβ) z, (j = 1, 2; N) Theorem 63 : If f 1 (z) = 1 z a n,1z n D a,c,λ [, α, A, B] and f 2 (z) = 1 z a n,2z n D a,c,λ [, α, A, B] with a n,2 1, n = 1, 2,,, then (f 1 f 2 )(z) D a,c,λ [, α, A, B] roof : Since = [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n,1 a n,2 (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n,1 a n,2, (1 2λ)(B A)( α) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n,1, (1 2λ)(B A)( α) 1 By Theorem 21, it follows that (f 1 f 2 )(z) D a,c,λ [, α, A, B] This comletes the roof of the theorem Corollary 65 : If f 1 (z) = 1 z a n,1z n D a,c,λ [, α, A, B] and f 2 (z) = 1 z a n,2z n D a,c,λ [, α, A, B] with (0 a n,2 1, n = 1, 2,, ), then (f 1 f 2 )(z) D a,c,λ [, α, A, B] Theorem 64 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class D a,c,λ [, α, A, B] and 2( αβ) a c ( α)[3a B 4λ(B A)] 0,

17 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 72 then the function h(z) defined by belongs to the class D a,c,λ [, α, A, B] h(z) = 1 z (a 2 n,1 a 2 n,2)z n, (20) roof : Since f 1 (z) D a,c,λ [, α, A, B], we get: and so [1 λ( n )][(1 B)n 2 2αB (B A)( α)] a n,1 1 (1 2λ)(B A)( α) [ [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) Similarly, since f 2 (z) D a,c,λ [, α, A, B], we have [ [1 λ( n)][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) Hence 1 2 [ [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) ] 2 a 2 n,1 1 ] 2 a 2 n,2 1 ] 2 (a 2 n,2 a 2 n,2) 1 In view of Theorem 21, it is sufficient to show that [ ] [1 λ( n)][(1 B)n 2 2αB (B A)( α)] (a 2 n,1 a 2 (1 2λ)(B A)( α) (c) n,2) 1 n1 (21) Thus the inequality (21) will be satisfied if, for n = 0, 1, 2, [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) or if 1 2 [ [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (1 2λ)(B A)( α) ] 2, [1λ( n )] [(1B)n22αB(BA)( α)] 2(1 2λ)(B A)( α) 0, (22),

18 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 73 for n = 0, 1, 2, Then left hand side of (22) is an increasing function of n, and hence (22) is satisfied for all n if 2( αβ) a c ( α)[3a B 4λ(B A)] 0, which is true by our assumtion This comletes the roof of the theorem Putting λ = 0 in Theorem 64, we obtain: Corollary 66 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class D a,c [, α, A, B] and 2( αβ) c ( α)(3a B) 0, a then the function h(z) defined by (20) belongs to the class D a,c [, α, A, B] Theorem 65 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class D a,c,λ [, α, A, B] Then the function h(z) defined by (20) belongs to the class D a,c,λ [, γ, A, B], where { 4c 2 (1 2λ)(1 B)(B A)( α) 2 γ 1 a 2 [2 2αB (B A)( α)] 2 ) 2(1 2λ)[(B A)( α)] 2 The result is shar for the functions f j (z)(j = 1, 2) defined by (19) roof : Noting that [[1 λ( n )][(1 B)n 2 2αB (B A)( α)]]2 [ ] 2 a 2 [(1 2λ)(B A)( α)] 2 n,j ( ) [1 λ( n )][(1 B)n 2 2αB (B A)( α)] (a) 2 n1 a n,j 1 (j = 1, 2), (1 2λ)(B A)( α) for f j (z) D a,c,λ [, γ, A, B](j = 1, 2), we have [ 1 [1 λ( n 2 (1 2λ)(B A)( α) )][(1 B)n 2 2αB (B A)( α)] Therefore, we have to find the largest γ such that [(1 B)n 2 2γB (B A)( γ)] ( γ) ] 2 (a 2 n,1 a 2 n,2) 1

19 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 74 that is, that [1 λ( n )][(1 B)n 2 2αB (B A)( α)]2, (n 0) 2(1 2λ)(B A)( α) 2 γ 2(2 n)(1 2λ)(1 B)(B A)( α) 2 [1 λ( n)] (n 0) ] 2 [(1 B)n 2 2αB (B A)( α)] 2 2(1 2λ)[(B A)( α)] 2 Now, defining the function ϕ(n) by ϕ(n) = 2(2 n)(1 2λ)(1 B)(B A)( α) 2 [1 λ( n)] (n 0) ] 2 [(1 B)n 2 2αB (B A)( α)] 2 2(1 2λ)[(B A)( α)] 2 We see that ϕ(n) is an increasing function of nthus, we conclude that { 4c(1 2λ)(1 B)(B A)( α) 2 γ ϕ(0) = 1 a[2 2αB (B A)( α)] 2 2(1 2λ)[(B A)( α)] 2 Hence the roof is comlete Putting A = β and B = β(0 < β 1) and λ = 0 in Theorem 65, we obtain: Corollary 67 : Let the functions f j (z)(j = 1, 2) defined by (16) be in the class [α, β] Then the function h(z) defined by (20) belongs to the class [γ, β], where 2β(1 β)( α)2 γ {1 ( αβ) 2 2β 2 ( α) The result is shar for the functions f j (z)(j = 1, 2) defined by (19) 7 Integral Transforms In this section, we consider integral transforms of functions in the classes [α, β] and [α, β] Theorem 71 : If f(z) is in the class [α, β], then the integral transforms F d 1 (z) = d 1 0 u d 1 f(uz)du, 0 < d <, (23)

20 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 75 are in the class [δ], 0 δ <, where { (d 2)( αβ) a c δ = δ(α, β, a, c, d,, λ) = (d 2)( αβ) a c The result is the best ossible for the function f(z) given by f(z) = 1 z β( α)d(1 2λ) β( α)d(1 2λ) (1 2λ)β( α) c αβ a z, ( N) roof : Suose f(z) = 1 z a nz n [α, β] Then we have F d 1 (z) = d 1 0 u d 1 f(uz)du = 1 z In view of Corollary 22, it is sufficient to show that Since f(z) [α, β], we have Thus (24) will be satisfied if or n δ δ da n n d 2 da n n d 2 zn [1 λ( n )][(1 B)n 2 αβ)] a n 1 (1 2λ)2β( α) 1 (24) (n δ)d ( δ)(n d 2) [1 λ( n )][(1 B)n 2( αββ)] (n 0), (1 2λ)2β( α) δ (n d 2)[1 λ( n)] [(1 B)n 2( αβ)] (n )d(1 2λ)2β( α)] (n d 2)[1 λ( n)]( )[(1 B)n 2( αβ) d(1 2λ)2β( α) (25) Since the right hand side of (25) is an increasing function of n, utting n = 0 in (25), we get δ Hence the roof is comlete { (d 2)( αβ)( a) β( α)d(1 2λ) c (d 2)( αβ) β( α)d(1 2λ)

21 International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 76 Putting λ = 0 in Theorem 71, we get: Corollary 71 : If f(z) is in the class [α, β], then the integral transforms (23)are in the class [δ], 0 δ <, where { (d 2)( αβ) a c δ = δ(α, β, a, c, d, ) = (d 2)( αβ) a c The result is the best ossible for the function f(z) given by f(z) = 1 β( α) z αβ c a z, ( N) β( α)d β( α)d

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