A SUBORDINATION THEOREM FOR SPIRALLIKE FUNCTIONS

We prove a subordination relation for a subclass of the class of λ-spirallike functions.


Introduction.
Let K denote the usual class of convex functions.Denote by S p (λ), −π/2 < λ < π/2, the class of functions f (z) = z + a 2 z 2 + ••• which are analytic in E and satisfy therein the condition Re e iλ zf (z) f (z) > 0. (1.1) Spacek [3] proved that members of S p (λ), known as λ spirallike functions, are univalent in E. In 1989, Silverman [2] proved that if then the function f (z) = z + ∞ n=2 a n z n belongs to S p (λ).Let us denote by G(λ), the class of function f (z) = z + ∞ n=2 a n z n whose coefficients satisfy the condition (1.2).Note that G(0) is a subclass of the class of starlike functions (with respect to the origin) (see Silverman [1]).
In this paper, we prove a subordination theorem for the class G(λ).To state and prove our main result we need the following definitions and lemma.
are analytic in |z| < r , then their Hadamard product/convolution, f * g is the function defined by the power series (1. 3) The function f * g is also analytic in |z| < r .
Definition 1.2.Let f be analytic in E, g analytic and univalent in E and f (0) = g(0).Then by the symbol f (z) ≺ g(z) (f subordinate to g) in E, we shall mean that f (E) ⊂ g(E).

Definition 1.3. A sequence {b n } ∞
1 of complex numbers is said to be a subordinating factor sequence if whenever (1.4) Lemma 1.4.The sequence {b n } ∞ 1 is a subordinating factor sequence if and only if This lemma which gives a beautiful characterisation of a subordinating factor sequence is due to Wilf [4].

Main theorem
for every function g in the class K.
then for every function g in K, we have a n z n be any member of the class G(λ) and let g(z) = z + ∞ n=2 c n z n be any function in the class K. Then (2.5) Thus, by Definition 1.3, the assertion of our theorem will hold if the sequence is a subordinating factor sequence, with a 1 = 1.In view of the lemma, this will be the case if and only if ( .8) Thus (2.7) holds true in E. This proves the first assertion.That Re f (z) > −(2 + sec λ)/(1+sec λ) for f ∈ G(λ) follows by taking g(z) = z/(1−z) in (2.1).To prove the sharpness of the constant (1+sec λ)/2(2+sec λ), we consider the function f 0 defined by f 0 (z) = z −(1/(1+sec λ))z 2 (|λ| < π/2), which is a member of the class G(λ).Thus from the relation (2.1) we obtain 1 + sec λ 2(2 + sec λ) f 0 (z) the constant (1 + sec λ)/2(2 + sec λ) is best possible.