On certain subclasses of multivalently meromorphic close-to-convex maps

Lemma 1. Let q be a convex univalent function in E and Re c > 0. Let h(z) = q(z) + p + 1 c zq′(z), where p is a positive integer. If p(z) = 1 + ap+1zp+1 + . . . is analytic in E and p(z) + 1 c zp′(z) ≺h(z), then p(z) ≺q(z) and q is the best dominant. P r o o f. We can assume that q is analytic and convex on E without any loss of generality, because otherwise we replace q(z) by qr(z) = q(rz), 0 < r < 1. These functions satisfy the conditions of the lemma on E. We can prove that pr(z) ≺qr(z), whic

Lemma 2. Let (3) w = (p + 1)2 + |c|2 −|(p + 1)2 −c2| 4(p + 1) Re c , Re c > 0. If h is analytic in E with h(0) = 1 and (4) Re  1 + zh′′(z) h′(z)  > −w, and if p(z) = 1 + ap+1zp+1 + . . . is analytic in E and satisfies

Theorem 1. Let q be a convex analytic function in E with q(0) = 1 and let h(z) = q(z) + (p + 1)zq′(z) n + 1 , n a positive integer. If f ∈Mp and Dnf(z) = 1 zp(1 −z)n+1 ∗f(z), then −zp+1 p (Dn+1f)′ ≺h ⇒−zp+1 p

Theorem 2. Let h be analytic in E with h(0) = 1, Re  1 + zh′′(z) h′(z)  > −w,

Corollary 1. Ln+1,p(α) ⊂Ln,p(r) for α < 1, where the best possible value of r is given by r = r(α, n) = 2α −1 + 2(1 −α) p + 1 (n + 1) 1 \ 0 t(n−p)/(p+1) 1 + t dt > α. P r o o f. Choose h(z) = 1 + z(2α −1) 1 + z in Theorem 2. Then h is convex and

Theorem 3. Let h be defined on E by h(z) = q(z) + p + 1 c −p + 1zq′(z),

Theorem 4. Let w = (p + 1)2 + |c′|2 −|(p + 1)2 −c′2| 4(p + 1) Re c′ , Re c′ > 0, c′ = c −p + 1. Let h be analytic in E and satisfy h(0) = 1, Re  1 + zh′′(z) h′(z)  > −w.

Theorem 5. Let f ∈Mp and let Ic(f) be defined by (17). Let α < 1. If Re  −zp+1 p (Dnf)′  > α −(1 −α) Re 1 c −p + 1 then Ic(f) ∈Ln,p(α). P r o o f. Denote Ic(f) by F and put (27) −zp+1(DnF(z))′ p

Definition 1. If f and g are analytic in E and g is univalent in E, then f is said to be subordinate to g, written f ≺g, if f(0) = g(0) and f(E) ⊂g(E).

Definition 2. Let z ∈E, t ≥0. A function L(z, t) is called a subor- dination chain if L(·, t) is analytic and univalent on E for all t ≥0, L(z, ·) is continuously differentiable on [0, ∞) for each z ∈E, and L(z, s) ≺L(z, l) for 0 ≤s < l.

Definition 3. Let H(p(z), zp′(z)) ≺h(z) be a first order differen- tial subordination. Then a univalent function q is called its dominant if p ≺q for all analytic functions p that satisfy the differential subordination. A dominant q is called the best dominant if q ≺q for all dominants q. For the general theory of differential subordination and its applications we refer to [5].