EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Lemma 1. ([6]) For a function p (z) ∈P of the form (6), the sharp inequality |pn| ⩽2 holds for each n ⩾1. Equality holds for the function p (z) = 1+z 1−z.

Lemma 2. ([7]) Let p (z) ∈P be a function of the form (6) and µ ∈C. Then |pn −µpkpn−k| ⩽2max {1, |2µ −1|} , 1 ⩽k ⩽n −1. If |2µ −1| ⩾1, then the inequality is sharp for the function p (z) = 1+z 1−z or its rotations. If |2µ −1| < 1, then the inequality is sharp for the function p (z) = 1+zn 1−zn or its rotations.

Lemma 3. ([10]) Let p (z) ∈P be a function of the form (6) and α, β, γ ∈ℜ. Then αp13 −βp1p2 + γp3 ⩽2 |α| + 2 |β −2α| + 2 |α −β + γ| . 3. Main results This section is devoted to the proof of our main results. We will now determine the coefficient estimates for functions belonging to SSC∗(ez), followed by logarithmic coeffi- cients of inverse functions and the second Hankel and Toeplitz determinants of logarithmic coefficients of inverse functions for the new subclass SSC∗(ez), as follows: 3.1. Co

Theorem 1. Let f (z) ∈SSC∗(ez) . Then |a2| ≤1 2, |a3| ≤1 2, |a4| ≤25 96, and |a5| ≤7 24.

Theorem 2. Let f (z) ∈SSC∗(ez) . Then |Γ1| ≤1 4, |Γ2| ≤1 4, |Γ3| ≤41 192, and |Γ4| ≤197 256.

Lemma 3, respectively.

Theorem 3. Let f (z) ∈SSC∗(ez) . Then H2,1 Γf−1
 ≤95 768.

Theorem 4. Let f (z) ∈SSC∗(ez) . Then H2,2 Γf−1
 ≤7691 36864.

Theorem 5. Let f (z) ∈SSC∗(ez) . Then T2,1 Γf−1
 ≤9 32.

Theorem 6. Let f (z) ∈SSC∗(ez) . Then T2,2 Γf−1
 ≤7165 9216.

Definition 1. Let SSC∗(ez) be the class of functions defined by zf′ (z) h (z) ≺ϕ (z) , z ∈E, where ϕ (z) = ez, is an analytic univalent function and h (z) = f(z)−f(−z) 2 .