On Janowski Starlike Functions

Lemma 2.1. Let w(z) be a nonconstant analytic in U with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r at a point z1, then one has z1w′z1  = kw z1 , (2.1) where k is real and k ≥1.

Lemma 2.2. Let f (z) ∈ and g(z) = z + ∞  n=2 bnzn ∈. (2.2) Then, the following fractional differential equation: Dλ z f (z) = 1 Γ(2 −λ)z−λg(z) (λ̸=2,3,4,...) (2.3) has the solution

Theorem 2.3. If f (z) ∈ satisfies the condition  z Dλ f (z) ′ Dλ f (z) −1  ≺ ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ (A −B)z 1 +Bz = F1(z),

Corollary 2.4. If f (z) ∈∗ λ (A,B), then Γ(2 −λ)zλ−1Dλ z f (z) B/(A−B) −1 < 1, B̸=0, log Γ(2 −λ)zλ−1Dλ z f (z) 1/A < 1, B = 0. (2.13)

Theorem 2.5. If f (z) ∈∗ λ (A,B), then 1 Γ(2 −λ)r1−λ(1 −Br)(A−B)/B ≤ Dλ z f (z) ≤ 1 Γ(2 −λ)r1−λ(1 +Br)(A−B)/B, B̸=0, 1 Γ(2 −λ)r1−λe−Ar ≤ Dλ z f (z) ≤

Corollary 2.6. Giving specific values to A and B, one obtains the distortion of the following class. (i) ∗ λ (1,−1), (ii) ∗ λ (1 −2β,−1), 0 ≤β < 1, (iii) ∗ λ (1,−1+ 1/M), M > 1/2, (iv) ∗ λ (β,−β), 0 ≤β < 1. Finally, we discuss the coefficient inequalities for f (z) ∈∗ λ (A,B).

Theorem 2.7. If f (z) ∈∗ λ (A,B), then an ≤ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ |A −B| (n −1)! Γ(n +1 −λ)