EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Lemma 1. ([9]) For a function p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn, z ∈E, the sharp inequality |pn| ⩽2 holds for each n ⩾1. Equality holds for the function p (z) = 1+z 1−z.

Lemma 2. ([10]) Let p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn 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. ([6]) Let p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn, z ∈E 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 upper bounds of the Taylor coefficients, logarithmic coefficients, and Hankel and Toeplitz determinants of logarithmic coefficients, respectively, as follows: 3.1. Taylor coefficients

Theorem 1. If f is of the form (1) belongs to S∗ SC (sin z) , then |a2| ⩽1 2, |a3| ⩽1 2, |a4| ⩽1 4, and |a5| ⩽1 2.

Theorem 2. If f is of the form (1) belongs to S∗ SC (sin z) , then |γ1| ⩽1 4, |γ2| ⩽1 4, |γ3| ⩽1 8, and |γ4| ⩽7 16.

Theorem 3. If f ∈S∗ SC (sin z) and has the series representation (1), then |H2,1 (Ff/2)| ⩽ 87 1024.

Theorem 4. If f ∈S∗ SC (sin z) and has the series representation (1), then |H2,2 (Ff/2)| ⩽33 256.

Theorem 5. If f ∈S∗ SC (sin z) , then γ12 −γ22 ⩽65 256.

Theorem 6. If f ∈S∗ SC (sin z) , then |T2,2 (γn)| ⩽11 32.

Definition 1. Let S∗ SC (sin z) be the class of functions defined by zf′ (z) h (z) ≺φ (z) , z ∈E, (2) where h (z) = f(z)−f(−z) 2 and φ (z) = 1 + sin z. It is observed that the classes S∗ SC and S∗