DIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR T. N. SHAMMUGAM, C. RAMACHANDRAN, M. DARUS S. SIVASUBRAMANIAN Abstract. By making use of the familiar Carlson Shaffer operator,the authors derive derive some subordination superordination results for certain normalied analytic functions in the open unit disk. Relevant connections of the results, which are presented in this paper, with various other known results are also pointed out. 1. Introduction Let H be the class of functions analytic in the open unit disk := { : < 1}. Let H[a, n] be the subclass of H consisting of functions of the form Let f() = a + a n n + a n+1 n+1 +. A m := { f H, f() = + a m+1 m+1 + a m+2 m+2 + } let A := A 1. With a view to recalling the principle of subordination between analytic functions, let the functions f g be analytic in. Then we say that the function f is subordinate to g if there exists a Schwar Received August 12, 2006. 2000 Mathematics Subject Classification. Primary 30C80; Secondary 30C45. Key words phrases. differential subordinations, differential superordinations, sominant, subordinant.
function ω, analytic in with such that We denote this subordination by ω(0) = 0 ω() < 1 ( ), f() = g(ω()) ( ). f g or f() g(). In particular, if the function g is univalent in, the above subordination is equivalent to f(0) = g(0) f( ) g( ). Let p, h H let φ(r, s, t; ) : C 3 C. If p φ(p(), p (), 2 p (); ) are univalent if p satisfies the second order superordination (1.1) h() φ(p(), p (), 2 p (); ), then p is a solution of the differential superordination (1.1). (If f is subordinate to F, then F is called to be superordinate to f.) An analytic function q is called a subordinant if q p for all p satisfying (1.1). An univalent subordinant q that satisfies q q for all subordinants q of (1.1) is said to be the best subordinant. Recently Miller Mocanu [6] obtained conditions on h, q φ for which the following implication holds: h() φ(p(), p (), 2 p (); ) q() p(). with the results of Miller Mocanu [6], Bulboacă [3] investigated certain classes of first order differential superordinations as well as superordination-preserving integral operators [2]. Ali et al. [1] used the results obtained by Bulboacă [3] gave sufficient conditions for certain normalied analytic functions f to satisfy q 1 () f () f() q 2 ()
where q 1 q 2 are given univalent functions in with q 1 (0) = 1 q 2 (0) = 1. Shanmugam et al. [7] obtained sufficient conditions for a normalied analytic functions f to satisfy q 1 () f() f () q 2() q 1 () 2 f () (f()) 2 q 2(). where q 1 q 2 are given univalent functions in with q 1 (0) = 1 q 2 (0) = 1. Recently, the first author combined with the third fourth authors of this paper obtained sufficient conditions for certain normalied analytic functions f to satisfy q 1 () L(a, c)f() q 2() where q 1 q 2 are given univalent functions with q 1 (0) = 1 q 2 (0) = 1 (see [8] for details; also see [9]). A detailed investigation of starlike functions of complex order convex functions of complex order using Briot- Bouquet differential subordination technique has been studied very recently by Srivastava Lashin [10]. Let the function ϕ(a, c; ) be given by ϕ(a, c; ) := n=0 (a) n (c) n n+1 (c 0, 1, 2,... ; ), where (x) n is the Pochhammer symbol defined by { 1, n = 0; (x) n := x(x + 1)(x + 2)... (x + n 1), n N := {1, 2, 3,... }.
Corresponding to the function ϕ(a, c; ), Carlson Shaffer [4] introduced a linear operator L(a, c), which is defined by the following Hadamard product (or convolution): (a) n L(a, c)f() := ϕ(a, c; ) f() = a n n+1. (c) n We note that L(a, a)f() = f(), L(2, 1)f() = f (), L(δ + 1, 1)f() = D δ f(), where D δ f is the Ruscheweyh derivative of f. The main object of the present sequel to the aforementioned works is to apply a method based on the differential subordination in order to derive several subordination results involving the Carlson Shaffer Operator. Furthermore, we obtain the previous results of Srivastava Lashin [10] as special cases of some of the results presented here. n=0 2. Preliminaries In order to prove our subordination superordination results, we make use of the following known results. Definition 1. [6, Definition 2, p. 817] Denote by Q the set of all functions f that are analytic injective on E(f), where E(f) = {ζ : lim ζ f() = }, are such that f (ζ) 0 for ζ E(f). Theorem 1. [5, Theorem 3.4h, p. 132] Let the function q be univalent in the open unit disk θ φ be analytic in a domain D containing q( ) with φ(w) 0 when w q( ). Set Q() = q ()φ(q()), h() = θ(q()) + Q(). Suppose that 1. Q is starlike univalent in,
If 2. R ( h ) () > 0 for. Q() then p() q() q is the best dominant. θ(p()) + p ()φ(p()) θ(q()) + q ()φ(q()), Theorem 2. [3] Let the function q be univalent in the open unit disk ϑ ϕ be analytic in a domain D containing q( ). Suppose that 1. R ϑ (q()) > 0 for, ϕ(q()) 2. q ()ϕ(q()) is starlike univalent in. If p H[q(0), 1] Q, with p( ) D, ϑ(p()) + p ()ϕ(p()) is univalent in, then q() p() q is the best subordinant. ϑ(q()) + q ()ϕ(q()) ϑ(p()) + p ()ϕ(p()), 3. Subordination Superordination for Analytic Functions We begin by proving involving differential subordination between analytic functions. ( ) µ L(a + 1, c)f() Theorem 3. Let H let the function q() be analytic univalent in such that q() 0, ( ). Suppose that q () is starlike univalent in. Let q() { R 1 + ξ } 2δ q() + β β (q())2 q () q() + q () (3.1) q > 0 ()
(3.2) (α, δ, ξ, β C; β 0) [ ] µ [ L(a + 1, c)f() L(a + 1, c)f() Ψ(a, c, µ, ξ, β, δ, f)() := α + ξ + δ [ ] L(a + 2, c)f() + βµ(a + 2) L(a + 1, c)f() 1. If q satisfies the following subordination: then (3.3) q is the best dominant. Ψ(a, c, µ, ξ, β, δ, f)() α + ξq() + δ(q()) 2 + β q () q() (α, δ, ξ, β, µ C; µ 0; β 0), ( ) µ L(a + 1, c)f() q() (µ C; µ 0) Proof. Let the function p be defined by ( ) µ L(a + 1, c)f() p() := ( ; 0; f A), so that, by a straightforward computation, we have p [ () (L(a + 1, c)f()) = µ p() L(a + 1, c)f() ] 1. ] 2µ
By using the identity: we obtain By setting (L(a, c)f()) = (1 + a)l(a + 1, c)f() al(a, c)f(), p () p() [ = µ (a + 2) L(a + 2, c)f() L(a + 1, c)f() ] (a + 2). θ(ω) := α + ξω + δω 2 φ(ω) := β ω, it can be easily observed that θ is analytic in C, φ is analytic in C \ {0} that Also, by letting φ(ω) 0 (ω C \ {0}). Q() = q ()φ(q()) = β q () q() h() = θ(q()) + Q() = α + ξq() + δ(q()) 2 + β q () q(), we find that Q() is starlike univalent in that ( h ) { () R = R 1 + ξ } 2δ q() + Q() β β (q())2 q () q() + q () q () (α, δ, ξ, β C; β 0). The assertion (3.3) of Theorem 3 now follows by an application of Theorem 1. > 0,
For the choices q() = 1 + A, 1 B < A 1 q() = 1 + B the following results (Corollaries 1 2 below). Corollary 1. Assume that (3.1) holds. If f A, Ψ(a, c, µ, ξ, β, δ, f)() α + ξ 1 + A 1 + B + δ ( ) γ 1 +, 0 < γ 1, in Theorem 3, we get 1 ( ) 2 1 + A + 1 + B ( ; α, δ, ξ, β, µ C; µ 0; β 0), where Ψ(a, c, µ, ξ, β, δ, f) is as defined in (3.2), then ( ) µ L(a + 1, c)f() 1 + A 1 + B 1 + A 1 + B is the best dominant. Corollary 2. Assume that (3.1) holds. If f A, ( ) γ 1 + Ψ(a, c, µ, ξ, β, δ, f)() α + ξ + δ 1 (α, δ, ξ, β, µ C; µ 0; β 0) (µ C; µ 0) β(a B) (1 + A)(1 + B) ( ) 2γ 1 + + 2βγ 1 (1 2 ) where Ψ(a, c, µ, ξ, β, δ, f)() is as defined in (3.2), then ( ) µ ( ) γ L(a + 1, c)f() 1 + (µ C; µ 0) 1 ( ) γ 1 + is the best dominant. 1
For a special case q() = e µa, with µa < π, Theorem 3 readily yields the following. Corollary 3. Assume that (3.1) holds. If f A, Ψ(a, c, µ, ξ, β, δ, f)() α + ξ e µa +δ e 2µA +βaµ (α, δ, ξ, β, µ C; µ 0; β 0) where Ψ(a, c, µ, ξ, β, δ, f)() is as defined in (3.2), then ( ) µ L(a + 1, c)f() e µa (µ C, µ 0) e µa is the best dominant. 1 For a special case when q() = (1 ) 2b (b C\{0}), a = c = 1, δ = ξ = 0, µ = α = 1 β = 1 b, Theorem 3 reduces at once to the following known result obtained by Srivastava Lashin [10]. then Corollary 4. Let b be a non ero complex number. If f A, 1 (1 ) 2b is the best dominant. 1 + 1 b f () f () f () 1 + 1, 1 (1 ) 2b Next, by appealing to Theorem 2 of the preceding section, we prove Theorem 4 below.
Theorem 4. Let q be analytic univalent in such that q() 0 q () q() be starlike univalent in. Further, let us assume that (3.4) [ 2δ R β (q())2 + ξ ] β q() > 0, (δ, ξ, β C; β 0). If f A, Ψ(a, c, µ, ξ, β, δ, f) is univalent in, then implies (3.5) ( ) µ L(a + 1, c)f() 0 H[q(0), 1] Q, α + ξq() + δ(q()) 2 + β q () q() Ψ(a, c, µ, ξ, β, δ, f) ( ; α, δ, ξ, β, µ C; µ 0; β 0), ( ) µ L(a + 1, c)f() q() (µ C; µ 0) q is the best subordinant where Ψ(a, c, µ, ξ, β, δ, f)() is as defined in (3.2). Proof. By setting ϑ(w) := α + ξw + δw 2 ϕ(w) := β 1 w, it is easily observed that ϑ is analytic in C. Also, ϕ is analytic in C \ {0} that ϕ(w) 0, (w C \ {0}).
Since q is convex (univalent) function it follows that, [ R ϑ (q()) 2δ ϕ(q()) = R β (q())2 + ξ ] β q() (δ, ξ, β C; β 0). The assertion (3.5) of Theorem 4 follows by an application of Theorem 2. We remark here that Theorem 4 can easily be restated, for different choices of the function q. Combining Theorem 3 Theorem 4, we get the following swich theorem. Theorem 5. Let q 1 q 2 be univalent in such that q 1 () 0 q 2 () 0, ( ) with q 1() q 1 () q 1() q 1 () being starlike univalent. Suppose that q 1 satisfies (3.4) q 2 satisfies (3.1). If f A, ( ) µ L(a + 1, c)f() H[q(0), 1] Q Ψ(a, c, µ, ξ, β, δ, f)() > 0, is univalent in, then implies α + ξq 1 () + δ(q 1 ()) 2 + β q 1() q 1 () (α, δ, ξ, β, µ C; µ 0; β 0), Ψ(a, c, µ, ξ, β, δ, f)() α + ξq 2 () + δ(q 2 ()) 2 + β q 2() q 2 () ( ) µ L(a + 1, c)f() q 1 () q 2 () (µ C; µ 0)
q 1 q 2 are respectively the best subordinant the best dominant. Corollary 5. Let q 1 q 2 be univalent in such that q 1 () 0 q 2 () 0 ( ) with q 1() q 1 () q 1() q 1 () being starlike univalent. Suppose q 1 satisfies (3.4) q 2 satisfies (3.1). If f A, (f ) µ H[q(0), 1] Q let Ψ 1 (µ, ξ, β, δ, f) := α + ξ [f ()] µ + δ [f ()] 2µ + 3 2 βµf () f () is univalent in, then implies α + ξq 1 () + δ(q 1 ()) 2 + β q 1() q 1 () Ψ 1(µ, ξ, β, δ, f) α + ξq 2 () + δ(q 2 ()) 2 + β q 2() q 2 () (α, δ, ξ, β, µ C; µ 0; β 0), q 1 () (f ) µ q 2 () (µ C; µ 0) q 1 q 2 are respectively the best subordinant the best dominant. Acknowledgment. The work presented here was partially supported by SAGA Grant: STGL-012-2006, Academy of Sciences, Malaysia. The authors also would like to thank the referee for critical comments suggestions to improve the content of the paper.
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