On a Class of Non-Univalent functions Associated

Lemma 1 Let z ∈Dr = {z : |z|= r}, then for each 0 ≤r < 1 and α ∈(−π, π], we have P0(r) ≤Re P0(reiα) ≤P0(−r).

Theorem 1 Let f ∈FLP, then the following holds I. (Growth Theorem) For |z|= r < 1, let max |z|=r Re P0(z) = P0(−r) and min |z|=r Re P0(z) = P0(r), then for |z|= r < 1 the following sharp inequality holds r exp Z r 0 P0(t) t dt  ≤|f(z)|≤r exp Z r

Lemma 2 Suppose a < 3/2 and assume that ζ(η) is defined as follows: ζ = ζ(η) = log  √η √1 −η  with η = e−π√1−2a 1 + e−π√1−2a , then LP(D) satisfies the following inclusion D(a, ra) := {ω ∈C : |ω −a|< ra} ⊂ΩLP, where ra =  

Theorem 2 Suppose 0 ≤α < 1 and −1 < B < A ≤1, then for f ∈A, the sharp FLP−radii for the classes S∗p, S∗s , ∆∗, S∗ϱ, S∗ρ, S∗℘, BS∗(α), S∗α,e and S∗(A, B) (see Table 1 in Appendix) are respectively given by (i) RFLP (S∗p) = tanh2(π/4). (ii) RFLP (S∗s ) = π/6. (iii) RFLP (∆∗) = 5/12. (iv) RFLP (S∗ϱ) = (cosh−1(3/2))2.

Theorem 3 Let 0 ≤α < 1 and 0 ≤γ ≤γα, where γα = tanh2(π√1 −α/2 √ 2). If p ∈PLP, then p ∈Pα, i.e p(z) is a Carath´eodory function of order α, in the disc |z|< γα.

Corollary 1 Let 0 ≤α < 1 and 0 ≤γ ≤γα, where γα is as defined in Theorem 3. If f ∈FLP, then f(z) is starlike of order α in the disc |z|< γα. This result is sharp.

Theorem 4 Assume 0 < α ≤1, then the sharp S∗(1 + αz)−radius for the class FLP is the unique positive root rα = tanh2(π√α/2 √ 2) of the equation 2 log((1 + √r)/(1 −√r))
2 −απ2 = 0, (5) where α is the radius of the disc {ω : |ω −1|< α} .

Corollary 2 Let f ∈A belong to FLP, then the following radii are sharp for the class FLP, (see Fig. 4) (i) The S∗e −radius is r1 = tanh2(λπ), where λ = (1/2) p (e −1)/2e. (ii) The S∗s −radius is r2 = tanh2(π/λ), where λ = 2 √ 2 csc 1. (iii) The S∗ϱ−radius is r3 = tanh2(πλ/2), where λ = sin(1/2). (iv) The S∗℘−radius is r4 = tanh2(π/2 √ 2e). (v) The S∗ρ−radius is r5 = tanh2(π √ λ/2), where λ = (1/2) sinh−1 1.

Corollary 3 Suppose 0 ≤β < 1 and f ∈FLP, then sharp S∗(β)−radius is tanh2(π√β/2 √ 2).

Corollary 3, we obtain the sharp S∗α−radius for the class FLP, where S∗α =  f ∈A : zf′(z)/f(z) −1 < 1 −α

Corollary 4 Let η = √ 2 −1 and suppose f ∈FLP, then the following holds (see Fig. 5) (i) f ∈S∗ L in |z|< tanh2 π√η/2 √ 2
≈0.376 . . . . (ii) f ∈S∗ RL in |z|< tanh2(π 4p√2 η(1 −√2η)/2 √ 2) ≈0.283 . . . .

Lemma 3 If p ∈Pn[A, B], then for |z|= r p(z) −1 −ABr2n 1 −B2r2n ≤|A −B|rn 1 −B2r2n . Particularly, if p ∈Pn(α), then p(z) −1 + (1 −2α)r2n 1 −r2n ≤2(1 −α)rn 1 −r2n .

Theorem 5 Let −1 ≤A ≤1, and suppose f ∈FLP, then the sharp F−radius is given by RF(FLP) = 1 2A + 3 p A2 + 12A + 28 −(5 + A)  =: RF.

Corollary 5 Let f ∈FLP, then sharp F1−radius and F2−radius for the class FLP are respectively given as (i) RF1(FLP) = √ 17 −4 ≈0.123... (ii) RF2(FLP) = ( √ 41 −6)/5 ≈0.080...

Theorem 6 Let δ = (π√β −1/ √ 2), where 1 < β < 3/2, and suppose f ∈FLP, then M(β)−radius is rβ = 1 + 2 (cot δ)2 −2|sec δ/(tan2 δ)|.

Theorem 7 Let f ∈A and suppose that g ∈FLP. Further assume that f(z) is majorized by g(z) in D, i.e f(z) ≪g(z), then for |z|≤rm ≈0.4220 . . . , |f′(z)|≤|g′(z)|, where rm is the unique positive root of the following equation 2π2r −(1 −r2)(π2 −2(log((1 + √r)/(1 −√r)))2) = 0. (9)

Theorem 8 Let f ∈FLP, then f ∈Ωin |z|< rL ≈0.522 . . . is the smallest positive root of 4f0(r)(log((1 + √r)/(1 −√r)))2 = π2 and g0(z) = z  exp Z z 0 P0(−t) t dt  = z + 8 π2 z2 −

Lemma 4 [7, Lemma 1, p.470] Let ν(z) be a non-constant analytic function in D, such that ν(0) = 0. If |ν(z)| attains its maximum value on the circle |z|= r at a point z0, then z0ν′(z0) = kν(z0), where k is real and k ≥1.

Theorem 9 Suppose 0 ≤t ≤1 and let f ∈A satisfy the following differential inequality t  1 + zf′′(z) f′(z)  + (1 −t)zf′(z) f(z) −1 < 1 6(3 + 2t), z ∈D, (14) then f ∈FLP.

Corollary 6 Let f ∈A satisfy the following differential inequalities (i.) (zf′(z)/f(z) + zf′′(z)/f′(z)) −1 < 4/3, or (ii.) (zf′(z)/f(z)) −1 < 1/2, or (iii.) zf′′(z)/f′(z) < 5/6, then f ∈FLP. Conclusion In the present investigation, we introduce a class of analytic functions associated with certain parabolic region. In particular, we have considered a case when parabola is lying majorly in the left half plane and symmetric about real axis. The other cases