Mathematics & Statistics

Theorem 2.1. The function f defined by (1.2) is in the class S∗ Ne if and only if 1 z [ f(z) ∗z −Lz2 (1 −z)2 ] ̸= 0 (2.1) for all L = 3+3eiθ−e3iθ 3eiθ−e3iθ , where θ ∈[0, 2π] and also L = 1.

Theorem 2.2. A necessary and sufficient condition for the function f defined by (1.2) to be in the class S∗ Ne is that a1 − ∞ ∑ n=2 3 −3n + 3eiθ −e3iθ 3eiθ −e3iθ anzn−1 ̸= 0. (2.9)

Theorem 2.3. If the function f defined by (1.2) satisfies the following inequality ∞ ∑ n=2 (3n −1)|an| ≤2a1, (2.11) then f ∈S∗ Ne.

Theorem 2.4. The function f defined by (1.2) is in the class CNe if and only if 1 z [ f(z) ∗z + [1 −2L]z2 (1 −z)3 ] ̸= 0 (2.12) for all L = 3+3eiθ−e3iθ 3eiθ−e3iθ , where θ ∈[0, 2π], and also L = 1. Reasoning along the similar lines as the proof of the Theorem 2.2 and Theorem 2.3, we establish following results for the class CNe. We are omitting the details.

Theorem 2.5. A necessary and sufficient condition for the function f defined by (1.2) to be in the class CNe is that a1 − ∞ ∑ n=2 n3 −3n + 3eiθ −e3iθ 3eiθ −e3iθ anzn−1 ̸= 0. (2.13)

Theorem 2.6. If the function f defined by (1.2) satisfies the following inequality ∞ ∑ n=2 n(3n −1)|an| ≤2a1, (2.14) then f ∈CNe.

Theorem 3.1. Let the functions fi defined by (1.3) be in the class S∗ Ne for every i = 1, 2, ...m. Then the quasi-Hadamard product f1 ∗f2 ∗... ∗fm belongs to the class S(m−1)Ne.

Theorem 3.2. Let the functions fi defined by (1.3) be in the class CNe for every i = 1, 2, ...m. Then the quasi-Hadamard product f1 ∗f2 ∗...∗fm belongs to the class S(2m−1)Ne.

Theorem 3.3. Let the functions fi defined by (1.3) be in the class CNe for every i = 1, 2, ...m; and let the functions gj defined by (1.4) be in the class S∗ Ne for every j = 1, 2, ...s. Then the quasi-Hadamard product f1 ∗f2 ∗... ∗fm ∗g1 ∗g2 ∗... ∗gs belongs to the class S(2m+s−1)Ne.

Lemma 4.1. [27, Lemma 3, p. 254] Let P be the class of analytic functions having the Taylor series of the form p(z) = 1 + p1z + p2z2 + p3z3 + · · · (4.1) satisfying the condition Re(p(z)) > 0 (z ∈D). Then 2p2 = p2 1 + (4 −p2 1)ξ for some ξ ∈D.

Theorem 4.2. Let the function f ∈A be in the class S∗ Ne. Then the best possible bounds on third order Hermitian–Toeplitz are given by −1 4 ≤|T3,1(f)| ≤1.

Lemma 4.3. [1, Lemma 3, p. 66] Let the function p ∈P, 0 ≤β ≤1 and β(2β−1) ≤δ ≤β. Then |p3 −2βp1p2 + δp3 1| ≤2.

Lemma 4.4. [33, Lemma 2.3, p. 507] Let p ∈P. Then for all n, m ∈N, |µpnpm −pm+n| ≤ { 2, 0 ≤µ ≤1; 2|2µ −1|, elsewhere. If 0 < µ < 1, then the inequality is sharp for the function p(z) = (1 + zm+n)/(1 −zm+n). In the other cases, the inequality is sharp for the function p0(z) = (1 + z)/(1 −z).

Lemma 4.5. [20] Let p ∈P. Then, for any real number µ, the following holds: |µp3 −p3 1| ≤ { 2|µ −4|, µ ≤4 3; 2µ √ µ µ−1, µ > 4 3. The result is sharp. If µ ≤4 3, equality holds for the function p0(z) := (1 + z)/(1 −z), and if µ > 4

Theorem 4.6. Let the function f ∈A be in the class S∗ Ne. Then, (i) |H(1) 3 (f)| ≤0.925696, (ii) |H(2) 3 (f)| ≤1.6225, (iii) |H(3) 3 (f)| ≤1.34575.