DIFFERENTIAL SUBORDINATION ASSOCIATED WITH NEW GENERALIZED DERIVATIVE OPERATOR

ROMAI J., v.12, no.1(2016), 77 89 DIFFERENTIAL SUBORDINATION ASSOCIATED WITH NEW GENERALIZED DERIVATIVE OPERATOR Anessa Oshah, Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia anessa.oshah@yahoo.com, maslina@ukm.edu.my Abstract The object of this work is to give some of the properties of differential subordination of derivative operator λ 1,λ 2,l,d f () studied earlier by Oshah and Darus [4]. Keywords: analytic functions, differential operator, starlike functions, convex functions, subordination. 2010 MSC: 30C45. 1. INTRODUCTION Let A be the class of functions f of the form f () = + a n n, (1) which are analytic in the unit disk U = { : C, < 1}. A function f A is said to be starlike of order α(0 α < 1) if and only if Re ( ) f () f () > α. This class is denoted by S (α). Further, a function f A is said to be convex of order α(0 α < 1) if and only if Re ( f ) () f () > α, U. This class is denoted by K(α). Let H(U) be the space of holomorphic functions in U. For a C and n N we denoted by H[a, n] = { f H(U), f () = a + a n n + a n+1 n+1 +..., U }. If f and g are analytic functions in U, we say that f is subordinate to g if there exists the Schwar function w, analytic in U, with w(0) = 0 and w() < 1 such that f () = g(w()), U. We denote this subordination by f () g(). If g() is univalent in U, the subordination is equivalent to f (0) = g(0) and f (U) g(u). Let ψ : C 3 U C and h be univalent in U. If p is analytic in U and satisfies the (second-order) differential subordination n=2 ψ(p(), p (), 2 p (); ) h(), ( U), (2) 77

78 Anessa Oshah, Maslina Darus p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p q for all p satisfying (2). A dominant q that satisfies q q for all dominants q of (2) is said to be the best dominant of (2). The best dominant is unique up to a rotation of U. The Hadamard product (convolution) of two analytic functions f () = + a k k and g() = + b k k is defined by f () g() = + a k b k k, ( U). (3) To prove our main result, we need the following lemmas. Lemma 1.1. [5]. Let h be a convex function with h(0) = a and let γ C \ {0} be a complex number with Re{γ} 0. If p H[a; n] and p() + p () γ h(), ( U), where q() = p() q() h(), ( U), γ n γ/n The function q is convex and is the best dominant. 0 h(t)t (γ/n) 1 dt, ( U). Lemma 1.2. [6]. Let g be a convex function in U and let where α > 0 and n is a positive integer. If h() = g() + nαg (), p() = g(0) + p n n + p n+1 n+1 +, ( U), is analytic in U and and this result is sharp. p() + αp () h(), ( U), p() g(),

Differential subordination associated with new generalied derivative operator 79 Lemma 1.3. [7]. Let f A, if Re { f } () f > 1 () 2, 2 f (t)dt, ( U, 0), 0 belongs to the class of convex functions. Now, we state the following generalied derivative operator as follows [4]: [ l(1 + = + (λ1 + λ 2 )(k 1)) + d ] mc(n, k)ak k, (4) where m, n, d N 0 = {0, 1, 2,...}, λ 2 λ 1 0, l 0, and > 0, C(n, k) = (n + 1) k 1 /(1) k 1. Note that, (ν) k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by { Γ(ν + k) 1, k = 0, ν C\{0}; (ν) k = = (5) Γ(ν) ν(ν + 1)(ν + 2)...(ν + k 1), k N = {1, 2, 3,...}. In particular, by choosing certain values of n, m, λ 1, λ 2, l, and d, we have D 0,0 λ 1,λ 2,l,d f () = D0,m 0,0,1,0 f () = f (); D1,0 = f (). (6) Now, to prove our results, we need the following equality ()+1 = (l(1 λ 1 λ 2 ) + d)( ())+ where ϕ d λ 2 () is analytic function given by ϕ d λ 2 () = + 2. MAIN RESULTS l(λ 1 + λ 2 )( ()), (7) k, U. In this work, we will use the method of differential subordination to derive certain properties of generalied derivative operator λ 1,λ 2,l,d f (). Note that differential subordination has been studied by various authors, and here we follow similar works done by G. I. Oros [8] and G. Oros and G. I. Oros [9].

80 Anessa Oshah, Maslina Darus Definition 2.1. For m, n, d N 0, λ 2 λ 1 0, l 0, > 0 and 0 α < 1, let (α) denote the class of functions f A which satisfy the condition R n,m λ 1,λ 2,l,d (γ) denote the class of functions f A which satisfy the condi- Also, let K n,m tion λ 1,λ 2,l,d Re( ) > α, ( U). (8) Re( ()) > γ, ( U). (9) It is clear that the class R 0,1 λ 1,0,1,0 (α) R(λ 1, α) the class of functions f A satisfy studied by Ponnusamy [10] and others. Re(λ 1 f () + f ()) > α, ( U), Theorem 2.1. Let 1 + (2α 1) h() =, ( U), be convex in U, with h(0) = 1 and 0 α < 1. If m, n, d N 0, λ 2 λ 1 0, l 0, > 0, f A and the differential subordination holds, ( ()) q() (+1 ) h(), ( U), (10) = 2α 1 + ( 2(1 α)() l(λ 1 + λ 2 ) (l+d)/l(λ 1+λ 2 ) σ l(λ 1 + λ 2 ) ), (11) where σ is given by t x 1 σ(x) = dt, ( U). (12) 0 1 + t The function q is convex and is the best dominant. Proof. By differentiating (7), with respect to, we obtain ()(+1 ) = ()( ()) + Using (13) in (10), the differential subordination (10) becomes ( ()) + l(λ 1 + λ 2 ) l(λ 1 + λ 2 )( ()). (13) ( ()) h() = 1 + (2α 1). (14)

Differential subordination associated with new generalied derivative operator 81 Let p() = ( ()) ( () [l(1 + = + (λ1 + λ 2 ))(k 1) + d] m ) [] m+1 C(n, k)a k k = 1 + p 1 + p 2 2 +, (p H[1, 1], U). (15) Using (15) in (14), the differential subordination becomes By using Lemma 1.1, we have p() q() = = p() + l(λ 1 + λ 2 ) p () h() = l(λ 1 + λ 2 ) (l+d)/l(λ 1+λ 2 ) l(λ 1 + λ 2 ) (l+d)/l(λ 1+λ 2 ) = 2α 1 + 0 2(1 α)() l(λ 1 + λ 2 ) (l+d)/l(λ 1+λ 2 ) σ 0 1 + (2α 1). (16) h(t)t ((l+d)/l(λ 1+λ 2 )) 1 dt ( ) 1 + (2α 1)t t ((l+d)/l(λ 1+λ 2 )) 1 dt 1 + t ( ), l(λ 1 + λ 2 ) where σ is given by (12); and this gives ( ()) q() = 2α 1 + ( 2(1 α)() l(λ 1 + λ 2 ) (l+d)/l(λ 1+λ 2 ) σ The function q is convex and is the best dominant. The proof is complete. l(λ 1 + λ 2 ) ). Theorem 2.2. If m, n, d N 0, λ 2 λ 1 0, l 0, > 0 and 0 α < 1, we have R n,m+1 λ 1,λ 2,l,d(α) Kn,m λ 1,λ 2,l,d (γ), where ( ) 2(1 α)() γ = 2α 1 + σ, l(λ 1 + λ 2 ) l(λ 1 + λ 2 ) and σ is given by (12). Proof. Let f R n,m+1 λ 1,λ 2,l,d(α), from Definition 2.1 we have Re(+1 ) > α, ( U),

82 Anessa Oshah, Maslina Darus which is equivalent to Using Theorem 2.1, we have (+1 ) h() = ( ()) q() = 2α 1 + 1 + (2α 1). ( 2(1 α)() l(λ 1 + λ 2 ) (l+d)/l(λ 1+λ 2 ) σ l(λ 1 + λ 2 ) Since q is convex and q(u) is symmetric with respect to the real axis, we deduce that Re( ()) > Req(1) = γ = γ(α, λ 1, λ 2, l, d) ( ) 2(1 α)() = 2α 1 + σ, l(λ 1 + λ 2 ) l(λ 1 + λ 2 ) for which we deduce R n,m+1 λ 1,λ 2,l,d(α) Kn,m λ 1,λ 2,l,d(γ). This complete the proof. Theorem 2.3. Let q be a convex function in U, with q(0) = 1 and let h() = q() + l(λ 1 + λ 2 ) q (), ( U). If m, n, d N 0, λ 2 λ 1 0, l 0, > 0, and f A satisfies the differential subordination (+1 ) h(), (17) ( ()) q(), ( U), and the result is sharp. ). Proof. Using (15) in (13), the differential subordination (17) becomes p() + l(λ 1 + λ 2 ) Using Lemma 1.2, we obtain p () h() = q() + l(λ 1 + λ 2 ) q () ( U). (18) p() q(), ( U). Hence ( ()) q(), ( U), and the result is sharp. This completes the proof of the Theorem 2.3. Now, we will give a simple application for Theorem 2.3.

Differential subordination associated with new generalied derivative operator 83 Example 2.1. For m = 1, n = 0, λ 2 λ 1 0, d N 0, l 0, > 0, q() = ()/(1 ), f A and U, by applying Theorem 2.3, we have h() = 1 + l(λ ( ) 1 + λ 2 ) = ()(1 2 ) + 2l(λ 1 + λ 2 ) 1 ()(1 ) 2. By using the equality (7) we find that ()D 0,1 = (l(1 λ 1 λ 2 ) + d)( f () ϕ d λ 2 ()) + l(λ 1 + λ 2 )( f () ϕ d λ 2 ()). Now, () ( D 0,1 () ) ( ) 2 = (l(1 λ 1 + λ 2 ) + d) + a k k l(1 + λ 2 (k 1)) + d ( ) 2 + l(λ 1 + λ 2 ) + ka k k. A straightforward calculation gives the following: ( D 0,1 () ) = = 1 + ( ) ()(l(1 + (λ1 + λ 2 )(k 1) + d) ) 2 ka k k 1, (19) ( ) k()(l(1 + = + (λ1 + λ 2 )(k 1) + d) ) 2 a k k () 1 ( ( f () ϕ d λ 2 () ) ( + Similarly, using (7), we see that ( l(1 + (λ1 + λ 2 )(k 1) + d k ()D 0,2 = (l(1 λ 1 λ 2 ) + d)(d 0,1 ()) ()(D 0,2 ) = ()(D 0,1 ()) )a k k )) () 1. + l(λ 1 + λ 2 )(D 0,1 ()). + l(λ 1 + λ 2 )(D 0,1 ()).

84 Anessa Oshah, Maslina Darus By using (19) we obtain (D 0,1 ()) = We get ( ) ()(l(1 + (λ1 + λ 2 )(k 1)) + d) ) 2 k(k 1)a k k 2. ()(D 0,2 λ 1,λ 2,l,d f ()) ( ()(l(1 + = () 1 + (λ1 + λ 2 )(k 1)) + d) l(1 + λ 2 (k 1)) + d) 2 ( ()(l(1 + (λ1 + λ 2 )(k 1)) + d) + l(λ 1 + λ 2 ) ) 2 that is = (D 0,2 ) = 1 + ( ( f () ϕ d λ 2 () ) ( + ) ka k k 1 ) k(k 1)a k k 1, ( ) 2 l(1 + (λ1 + λ 2 )(k 1)) + d ka k k 1 ( (l(1 + (λ1 + λ 2 )(k 1)) + d) 2 k ()) From Theorem 2.3, we deduce that ( ( f () ϕ d λ 2 () ) ( + implies that ( ( f () ϕ d λ 2 () ) ( + ( (l(1 + (λ1 + λ 2 )(k 1)) + d) 2 k ()) ( l(1 + (λ1 + λ 2 )(k 1) + d k, ( U). 1 )a n n )) () 1. )a n n )) () 1 ()(1 2 ) + 2l(λ 1 + λ 2 ) ()(1 ) 2, )a k k )) () 1 Theorem 2.4. Let q be a convex function in U, with q(0) = 1 and let h() = q() + q (), ( U). If m, n, d N 0, λ 2 λ 1 0, l 0, > 0, and f A satisfies the differential subordination ( ) h(), (20)

Differential subordination associated with new generalied derivative operator 85 and the result is sharp. q(), ( U), Proof. Let p() = Dn,m, ( U). (21) Differentiating (21), with respect to, we obtain ( ) = p() + p (), ( U). (22) Using (21), the differential subordination (20), becomes Using Lemma 1.2, we deduce that and by using (21), we get This proves Theorem 2.4. p() + p () h() = q() + q (), ( U). p() q(), ( U), q(), ( U). Now, we will give a simple application for Theorem 2.4. Example 2.2. For n = 0, m = 1, λ 2 λ 1 0, d N 0, l 0, > 0,, q() = 1/(1 ), f A and U, from Theorem 2.4 we obtain h() = 1 ( ) 1 + 1 1 = 1 (1 ) 2. From Example 2.1, we have ()D 0,1 = (l(1 λ 1 λ 2 ) + d)( f () ϕ d λ 2 ()) + l(λ 1 + λ 2 )( f () ϕ d λ 2 ()). ()(D 0,1 ) = ()( f () ϕ d λ 2 ()) + l(λ 1 + λ 2 )( f () ϕ d λ 2 ()). From Theorem 2.4 we deduce that ( f () ϕ d λ 2 ()) + l(λ 1 + λ 2 ) ( f () ϕ d λ () 2 ) 1 (1 ) 2,

86 Anessa Oshah, Maslina Darus implies that (l(1 λ 1 λ 2 ) + d)( f () ϕ d λ 2 ()) + l(λ 1 + λ 2 )( f () ϕ d λ 2 ()) (l + b) 1 1. Theorem 2.5. Let h be a convex function in U, with h(0) = 1, 0 α < 1 and let h() = 1 + (2α 1), ( U). If m, n, d N 0, λ 2 λ 1 0, l 0, > 0 and f A satisfies the differential subordination ( ) h(), (23) 2(1 α) ln() q() = 2α 1 +, ( U). The function q is convex and is the best dominant. Proof. Let = 1 + p() = Dn,m ( l(1 + (λ1 + λ 2 )(k 1)) + d ) mc(n, k)ak k 1 = 1 + p 1 + p 2 2 +, (p H[1, 1], U). Differentiating (24) with respect to, we obtain (24) ( ) = p() + p (), ( U). (25) Using (25), the differential subordination (23) becomes p() + p () h() = Using Lemma 1.1, we deduce that 1 + (2α 1), ( U). p() q() = 1 h(t)dt 0 = 1 ( ) 1 + (2α 1)t dt 0 1 + t = 1 ( 1 dt + (2α 1) 0 1 + t 0 2(1 α) ln() = 2α 1 +. ) t 1 + t dt

Differential subordination associated with new generalied derivative operator 87 By using (24), we have q() = 2α 1 + The proof of Theorem 2.5 is complete. 2(1 α) ln(). Corollary 2.1. If f R n,m λ 1,λ 2,l,d(α), λ Re 1,λ 2,l,d f () > (2α 1) + 2(1 α) ln 2, ( U). Proof. Since f R n,m λ 1,λ 2,l,d(α), from the Definition 2.1, we have which is equivalent to Using Theorem 2.5, we obtain Re ( ) > α, ( U), ( D n,m ) 1 + (2α 1) h() =. q() = 2α 1 + 2(1 α) ln(). Since q is convex and q(u) is symmetric with respect to the real axis, we have that λ Re 1,λ 2,l,d f () > Re q(1) = (2α 1) + 2(1 α) ln 2, ( U), that proves the Corollary. Theorem 2.6. Let h H(U), with h(0) = 1, h (0) 0 which satisfies the inequality ( ) Re h () h > 1, ( U). () 2 If m, n, d N 0, λ 2 λ 1 0, l 0, > 0 and f A satisfies the differential subordination ( ) h(), (26) q() = 1 h(t)dt. 0

88 Anessa Oshah, Maslina Darus Proof. Let = 1 + p() = Dn,m ( l(1 + (λ1 + λ 2 )(k 1)) + d ) mc(n, k)ak k 1 = 1 + p 1 + p 2 2 +, (p H[1, 1], U). Differentiating (2.20), with respect to, we obtain (27) ( ) = p() + p (), ( U). (28) Using (28), the differential subordination (26), becomes Using Lemma 1.1, we deduce that by using (27), we get p() + p () h(), ( U). p() q() = 1 q() = 1 0 h(t)dt, 0 h(t)dt. From Lemma 1.3, we see that the function q is convex, and from Lemma 1.1, q is the best dominant for subordination (26). This completes the proof of Theorem 2.6. Remark 2.1. Some other works related to differential subordination can be found in [1], [2], [3]. Acknowledgements. The work here is supported by FRGS/1/2016/STG06/UKM/01/1. References [1] M. H. Al-Abbadi, M. Darus, Differential subordination for new generalised derivative operator, Acta Univ. Apulensis, 20 (2009), 265-280. [2] M. H. Al-Abbadi, M. Darus, Differential subordination defined by new generalised derivative operator for analytic functions, International Journal of Mathematics and Mathematical Sciences, Article ID 369078, 15 pages, 2010. [3] R. W. Ibrahim, M. Darus, S. Momani, Subordination and superordination for certain analytic functions containing fractional integral, Surv. Math. Appl. 4 (2009), 111 117. [4] A.Oshah, M.Darus, Differential sandwich theorems with a new generalied derivative operator, Advances in Mathematics: Scientific Journal, 3 (2014), 117-124. [5] D. J. Hallenbeck, S. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math., Soc., 52 (1975), 191-195.

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