Study on subclasses of analytic functions

Lemma 1 [19] Let f and g be starlike functions of order 1/2 then so is f ∗g .

Lemma 2 [20] Let P be a complex function in U with ℜ(P(z)) > 0 for z ∈U and let h be a convex function in U. If p is analytic in U with p(0) = h(0) and if p(z) + P(z)zp′(z) ≺h(z), then p(z) ≺h(z).

Lemma 3 [21] Let c > −1 and let Ic : A →A be the integral operator defined by F = Ic(f), where F(z) = c + 1 zc Z z 0 tc−1f(t)dt. Let h be a convex function, with h(0) = 1 and then ℜ(h(z) + c) > 0, z ∈U. If f ∈A and zf′(z) f(z) ≺h(z), then zF′(z) F(z) ≺q(z) ≺h(z), where q is univalent and satisfies the differential equation q(z) + zq′(z) q(z) + c = h(z).

Lemma 4 [22] Let f and g, respectively be in the classes K and S, then for every function F ∈A, we have (f(z) ∗g(z)F(z)) (f(z) ∗g(z)) ∈co(F(U)), z ∈U, where co denotes the closed convex hull.

Lemma 5 [22] Let f and g be univalent starlike of order 1 2 for every function F ∈A, we have (f(z) ∗g(z)F(z)) (f(z) ∗g(z)) ∈co(F(U)), z ∈U, where co denotes the closed convex hull.

Theorem 1 Let h be a convex function in U with h(0) = 1, h(z) = h(z) and let µ + λ ≥ν + β, if f ∈SVn,ν m,λ(α, β, µ)(h) then z(In,ν λ (α, β, µ)fm(z))′ In,ν λ (α, β, µ)fm(z) ≺h(z), z ∈U. (16) Moreover, if ℜ  h(z) + ν+β−µ−λ µ+λ  > 0 in U then

Corollary 1 Let ℜ h(z)
> 0, if f ∈SVn,ν m,λ(α, β, µ)(h) then In,ν λ (α, β, µ)fm ∈ S and hence In,ν λ (α, β, µ)f is close to convex function.

Theorem 2 Let ℜ h(z
> 0 and h(z) = h(z) then the following inclusions hold SVn+1,ν m,λ (α, β, µ)(h) ⊆SVn,ν m,λ(α, β, µ)(h) ⊆SVn−1,ν m,λ (α, β, µ)(h).

Theorem 4 implies f(z) ∈SVn,ν m,λ(α, β, µ)(h). Hence SVn+1,ν m,λ (α, β, µ)(h) ⊆SVn,ν m,λ(α, β, µ)(h). Similarly we can show that SVn,ν m,λ(α, β, µ)(h) ⊆SVn−1,ν m,λ (α, β, µ)(h), and SVn−1,ν m,λ (α, β, µ)(h) ⊆SVn−2,ν m,λ (α, β, µ)(h) and so on, therefore

Corollary 2 Taking h(z) = 1+z 1−z, in Theorem 2, then SVn+1,ν m,λ (α, β, µ)(1 + z 1 −z) ⊆SVn,ν m,λ(α, β, µ)(1 + z 1 −z) ⊆SVn−1,ν m,λ (α, β, µ)(1 + z 1 −z). Implies that SVn,ν m,λ(α, β, µ)(1+z 1−z) are starlike functions with respect to symmet- ric conjugate points.

Theorem 3 If f ∈SVn,ν m,λ(α, β, µ)(h) then f ∗g ∈SVn,ν m,λ(α, β, µ)(h) where ℜ h(z)
> 0 and g is a convex function with real coefficients in U.

Theorem 4 If Ωαf(z) ∈SVn,ν m,λ(α, β, µ)(h) then f(z) ∈SVn,ν m,λ(α, β, µ)(h), where ℜ h(z)
> 0, h(z) = h(z).

Theorem 5 Let 0 ≤α1 < α < 1, and Re(h(z)) > 1 2, then the following inclusions hold SVn,ν m,λ(α, β, µ)(h) ⊆SVn,ν m,λ(α1, β, µ)(h) .