Stud. Univ. Babe¸s-Bolyai Math. 71(2026), No. 2, 235–252

Lemma 2.1. Let d ∈P be of the form (2.1). (i) Then, for n ≥1 |dn| ≤2. (2.2) The inequality holds for all n ≥1 if and only if d(z) = 1 + λz 1 −λz , |λ| = 1. (ii) Also, if µ ≥0 then |dn+k −µdndk| ≤2 max {1; |2µ −1|} = ( 2, if 0 ≤µ ≤1, 2|2µ −1|, otherwise. (2.3)

Lemma 2.2. [3, Lemma 2.2.] If d ∈P has the form (2.1), then αd3 1 −βd1d2 + γd3 ≤2 (|α| + |β −2α| + |α −β + γ|) . (2.4)

Lemma 2.3. [1, Lemma 3, p. 66] Let d ∈P and has the expansion of the form (2.1). If B ∈[0, 1] with B(2B −1) ≤D < B, then d3 −2Bd1d2 + Dd3 1 ≤2. (2.5) 3. Coefficient estimates and Fekete-Szeg¨o inequality In this section, we aim to examine the upper bounds of the first four initial coefficients, along with the Fekete–Szeg¨o functional |a3 −µa2 2|, for the considered class Sr ∗. The focus is placed on deriving appropriate upper estimates for these coefficients within the framework of the class Sr

Theorem 3.1. Let the function f ∈A be of the form (1.1) belongs to the class Sr ∗. Then |a2| ≤1 + r 1 + 3r, |a3| ≤ 9 −5r 16(1 + 3r), |a4| ≤| (57r −5) (1 + r) | + (1 + r) (519r + 197) 192 (1 + 3r) (5r + 3) , |a5| ≤|227r2 + 10r −13| + 6349r2 + 3926r + 253 768 (1 + 3r) (5r + 3) + 1 4.

Theorem 3.2. If f ∈Sr ∗has the form (1.1), then for any complex number µ we have |a3 −µa2 2| ≤1 2 max ( 1;

Corollary 3.3. If f ∈Sr ∗has the form (1.1), then a3 −a2 2 = |H2,1(f)| ≤1 2 max ( 1;

Theorem 4.1. If the function f ∈A given by (1.1) belongs to the class Sr ∗, then |H2,2(f)| ≤44649r3 + 50245r2 + 16715r + 1927 1536 (1 + 3r)2 (5r + 3) . (4.1)

Theorem 4.2. If the function f ∈A given by (1.1) belongs to the class Sr ∗, then |a4 −a2a3| ≤(1 + r) [|303r2 + 144r −7| + 561r2 + 432r + 103
] 96 (1 + 3r)2 (5r + 3) . (4.4)

Theorem 5.1. If f ∈Sr ∗given by (1.1), then |γ1| ≤ (1 + r) 2(1 + 3r), |γ2| ≤1 4 max ( 1;

Theorem 6.1. If f ∈Sr ∗is given by (1.1) and its inverse f −1 has the form (1.6), then |h2| ≤1 + r 1 + 3r, (6.1) |h3| ≤1 2 max ( 1; 47r2 + 42r + 23 8 (1 + 3r)2 ) , (6.2) |h4| ≤(1 + r) 4905r3 + 5607r2 + 2583r + 633 + |741r3 + 1083r2 + 315r −59|

Theorem 7.1. If f ∈Sr ∗given by (1.1), then |Γ1| ≤ (1 + r) 2(1 + 3r), |Γ2| ≤1 4 max ( 1;

Definition 1.1. A function f ∈A given by (1.1) is said to be in the class Sr ∗if the below condition holds true: Sr ∗= ( f ∈S : (1 + r)zf ′(z) f(z) −rf(−z) ≺2√1 + z 1 + e−z ; z ∈D )

Definition 1.1 there exists an analytic function w with w(0) = 0 and |w(z)| < 1, (z ∈ D) such that (1 + r)zf ′(z) f(z) −rf(−z) = 2 p 1 + w(z) 1 + e−w(z) . (3.2) Writing the analytic function w in terms of d ∈P, that is d(z) = 1 + w(z)