Earthline Journal of Mathematical Sciences

Lemma 1.1. If w(z) ∈P, the class of functions which are analytic in U with ℜ(w(z)) > 0, (z ∈U) and have the form w(z) = 1 + w1z + w2z2 + w3z3 + · · · , (z ∈U); then |wn| ≤2 for each n ∈N. We use the m-fold symmetric function w in the class P (see [16]) of the form: w(z) = 1 + wmzm + w2mz2m + w3mz3m + · · · , (z ∈U). In the present investigation, with reference to the ⋋-pseudo-starlike function class defined by Babalola [3] and the work of Joshi and Yadav [10], we obtain estimates on the initial c

Theorem 2.2. Let g(z) given by (1.3) be in the class Λm Σ (η, ⋋, φ), 0 < φ ≤1. Then |bm+1| ≤ 2φ r (φ + 1)2 + φ(η2 + 2η(1 −m2 −m) −m + m(m + 1)⋋) + ⋋(m + 1)(⋋(m + 1) −2η −2) (2.3) and |b2m+1| ≤ 2φ (2m + 1) ⋋−2ηm −1 + 2φ2(m + 1) ((m + 1) ⋋−η −1)2 .

Theorem 3.2. Let g(z) given by (1.3) be in the class Λm Σ (η, ⋋, δ), 0 ≤δ < 1. Then, |bm+1| ≤ 2 √ 1 −δ r ⋋2(m + 1)2 + 2η2 −2(m + 1)η ⋋+ ⋋(m + 1)(m −2) −(m + 1)(2mη + 1) + 4η + 2 (3.3) and |b2m+1| ≤ 2(m + 1)(1 −δ)2 ((m + 1) ⋋−η −1)2 +

Corollary 3.3. Let g(z) given by (1.3) be in the class Λm Σ (⋋, φ), 0 < φ ≤1. Then |bm+1| ≤ 2φ p 1 + φ(m2 + m ⋋−m) + ⋋(m + 1)(⋋m + ⋋−2) and |b2m+1| ≤ 2φ (2m + 1) ⋋−1 + 2φ2(m + 1) ((m + 1) ⋋−1)2 .

Corollary 3.4. Let g(z) given by (1.3) be in the class Λm Σ (⋋, δ), 0 ≤δ < 1. Then, |bm+1| ≤ 2 √ 1 −δ p ⋋2(1 + m)2 + ⋋(m + 1)(m −2) −(m + 1) + 2 and |b2m+1| ≤2(m + 1)(1 −δ)2 ((m + 1) ⋋−1)2 + 2(1 −δ) (2m + 1) ⋋−1. http://www.earthlinepublishers.com

Corollary 3.7. Let f(z) given by (1.1) be in the class ΛΣ(η, ⋋, φ), 0 < φ ≤1. Then |b2| ≤ 2φ p [(η + 1)2 + φ(η2 −2η + 2 ⋋−1) + 4 ⋋(⋋−η −1)] and |b3| ≤ 2φ 3 ⋋−2η −1 + 4φ2 (2 ⋋−η −1)2 .

Corollary 3.8. Let f(z) given by (1.1) be in the class ΛΣ(η, ⋋, δ), 0 ≤δ < 1. Then, |b2| ≤ s 2(1 −δ) η2 + 2 ⋋2 −2 ⋋η −⋋ Earthline J. Math. Sci. Vol. 6 No. 2 (2021), 209-223

Corollary 3.9. [9] Let f(z) given by (1.1) be in the class ΛΣ(⋋, φ), 0 < φ ≤1. Then |b2| ≤ 2φ p 1 + φ(2 ⋋−1) + 4 ⋋(⋋−1) and |b3| ≤ 2φ 3 ⋋−1 + 4φ2 (2 ⋋−1)2 .

Corollary 3.10. [9] Let f(z) given by (1.1) be in the class ΛΣ(⋋, δ), 0 ≤δ < 1. Then, |b2| ≤ r 2(1 −δ) 2 ⋋2 −⋋ and |b3| ≤4(1 −δ)2 (2 ⋋−1)2 + 2(1 −δ) 3 ⋋−1 . Acknowledgement The authors wish to express their sincere thanks to the referees of this paper for valuable comments and suggestions. References [1] R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam, Coefficient estimates for