New Results on the Chebyshev Polynomial Bounds for Classes of Univalent Functions

Theorem 2.1. [4] Let the function f(z) given by (1.1) be in the class K(λ, t). Then (2.4) |a2| ≤ 2t 1 + λ and (2.5) |a3| ≤(2λ2 + 10λ + 4)t2 (1 + 2λ)(1 + λ)2 + t 1 + 2λ − 1 2(1 + 2λ). However, since the problem of estimating coefficients |an| for n ≥2 is still an open problem, the authors in this present work are interested in the higher coefficient bounds for the class

Lemma 3.1. Let the function f(z) given by (1.1) be in the class K(λ, t) then (3.1) a2 = 2tC1 1 + λ, a3 = 2tC2 2(1 + 2λ) + 1 2(1 + 2λ){(4t2 −1) + 1 + 3λ (1 + λ)24t}C2 1.

Theorem 3.1. Let the function f(z) given by (1.1) be in the class K(λ, t). Then |a4| ≤(16λ4 + 92λ3 + 208λ2 + 100λ + 8)t3 3(3λ + 1)(2λ + 1)(λ + 1)3 + (16λ2 + 58λ + 14)t2 3(3λ + 1)(2λ + 1)(λ + 1) + (4λ2 + 23λ + 5)t 3(3λ + 1)(2λ + 1)(λ + 1) − 2 3(3λ + 1) and |a5| ≤(144λ7 + 736λ6 + 2442λ5 + 3310λ4 + 1778λ4 + 478λ2 −142λ −82)t4 3(4λ + 1)(3λ + 1)(2λ + 1)2(λ + 1)4 + (216λ5 + 1274λ4 + 2934λ3 + 1389λ2 + 771λ + 31)t3

Theorem 4.1. Let f given by (1.1) be in the class K(λ, t), then |a2a4 −a2 3| ≤  1 + 2λ 3(1 + λ)4[(1 + 2λ)(1 + 3λ)]2[(1 + λ)3(1 + 3λ)(23 −λ) −8(λ)2(1 + 3λ)(55λ + 22) + 8λ(1 + 2λ) + 4(1 + λ)(1 + 3λ)(33 + 16λ)]t4 + t5 + 4[(1 + 2λ)(5 + λ) + (5λ −3)] (1 + λ)2(1 + 2λ)(1 + 3λ) t3  −  1 3(1 + λ)4[(1 + 2λ)(1 + 3λ)]2[13(1 + 2λ)2(1 + 3λ)

Corollary 4.1. If f ∈K(t), then |a4| ≤3 4t3 + 11 9 t2 + 4 9t −1 6, |a5| ≤3733 4320t4 + 441 144t3 −1237 2160t2 −621 1440t −3 40, |a2a4 −a2 3| ≤71 72t2 + 1297