A Class of Analytic Functions associated with Sine Hyperbolic Functions

Theorem 1. Let f ∈Gsh and f0 be given as in (1.5). Then for |z| = r, we have (1) (growth theorem) −f0(−r) ≤|f(z)| ≤f0(r). (2) (covering theorem) either f is a rotation of f0 or {w ∈C : |w| ≤−f0(−1)} ⊂f(D), where −f0(−1) = limr→1(−f0(−r)). (3) ℜf(z) z ≤f0(r) r and |f ′(z)| ≤(1 + sinh r)f0(r) r . In this paper, we consider some important properties like convolution problems, necessary and sufficient conditions, coefficient problems, convex combination, upper bounds for coefficients, Fekete-szeg¨o proble

Lemma 1. [18]. If k ∈P and it is of the form (1.2), then for λ ∈C (2.1) |cn| ≤2 for n ≥1, and (2.2) c2 −λc2 1 ≤2 max {1; |2λ −1|} .

Lemma 2. [18]. If k ∈P and is represented by (1.2), then c2 −νc2 1 ≤    −4ν + 2 (ν ≤0), 2 (0 ≤ν ≤1), 4ν −2 (ν ≥1).

Lemma 3. [14, 15] If k ∈P be expressed in series expansion (1.2), then 2c2 = c2 1 + x 4 −c2 1
for some x, |x| ≤1 and 4c3 = c3 1 + 2 4 −c2 1
c1x − 4 −c2 1

Lemma 4. If k ∈P be expressed in series expansion (1.2), then (2.3) ac3 1 −bc1c2 + dc3 ≤2 |a| + 2 |b −2a| + 2 |a −b + d|

Lemma 5. [21] Let m, n, l and r satisfy the inequalities 0 < m < 1, 0 < r < 1 and 8r (1 −r)  (mn −2l)2 + (m (r + m) −n)2 + m (1 −m) (n −2rm)2 ≤4m2 (1 −m)2 r (1 −r) . If k ∈P and has power series (1.2) then lc4 1 + rc2 2 + 2mc1c3 −3 2nc2 1c2 −c4 ≤2.

Lemma 6. [8] Let w (z) be analytic in D with w (0) = 0. If |w (z)| attains its maximum value on the circle |z| = r at a point z0 = reiθ, for θ ∈[−π, π] , we can write that z0w′ (z0) = mw (z0) , where m is real and m ≥1. 3. Main Results We begin with the following result:

Theorem 2. Let f ∈A be of the form (1.1). Then f ∈Gsh, if and only if (3.1) 1 z

Theorem 3. Let f ∈A be of the form (1.1). Then necessary and sufficient condi- tion for function f (z) belong to class Gsh is that (3.2) 1 − ∞ X n=2 n − 1 + sinh eiθ

sinh (eiθ) anzn−1 ̸= 0.

Theorem 4. Let f ∈A and satisfies (3.3) ∞ X n=2

Theorem 5. The class Gsh is convex.

Theorem 6. Let f ∈Gsh be of the form (1.1). Then |a2| ≤ 1, |a3| ≤ 1 2, |a4| ≤ 1 3, |a5| ≤ 1

Corollary 1. Let f ∈Gsh be of the form (1.1). Then a3 −a2 2 ≤1 2. The result is sharp.

Theorem 7. Let f ∈Gsh be of the form (1.1). Then a2a4 −a2 3 ≤1 36. The result is sharp.

Theorem 8. Let f ∈Gsh be of the form (1.1). Then |H3,1 (f)| ≤1 4 ≃0.25.

Theorem 9. For −1 ≤B < A ≤1 and f ∈A, if the conditions (4.1) |α| ≥ A −B 1 + cos 1 −sin 1 −|B| (1 + sinh 1 + cosh 1) and 1 + αzf ′ (z) ≺1 + Az 1 + Bz , holds. Then f (z) z ≺1 + sinh z.

Corollary 2. For −1 ≤B < A ≤1, and g ∈A, then if the following conditions |α| ≥ A −B 1 + cos 1 −sin 1 −|B| (1 + sinh 1 + cosh 1) and (4.6) 1 + αz2g′ (z) g (z)  2 + zg′′ (z) g′ (z) −zg′ (z) g (z)  ≺1 + Az 1 + Bz ,

Theorem 10. For −1 ≤B < A ≤1 and f ∈A, then if the conditions (4.7) |α| ≥ (A −B) (1 + sinh 1) 1 + cos 1 −sin 1 −B (1 + cosh 1 + sinh 1) and (4.8) 1 + αzf ′ (z) f (z) ≺1 + Az 1 + Bz , holds, then f (z) z ≺1 + sinh z.

Corollary 3. For −1 ≤B < A ≤1, and f ∈Ap then if the condition |α| ≥ (A −B) (1 + sinh 1) 1 + cos 1 −sin 1 −B (1 + cosh 1 + sinh 1), and 1 + α  2 + zg′′ (z) g′ (z) −zg′ (z) g (z)  ≺1 + Az 1 + Bz , holds then g ∈Gsh.

Theorem 11. For −1 ≤B < A ≤1 and f ∈Mp then if the condition (4.9) |α| ≥ (A −B) (1 + sinh (1))2 1 + cos (1) −sin (1) −B (1 + cosh (1) + sinh (1)) is true and (4.10) 1 + αz2f ′ (z) (f (z))2 ≺1 + Az 1 + Bz , then f (z) z ≺ √

Lemma 6 for m ≥1, we have, z0w′ (z0) = mw (z0) . So we have

Corollary 4. For −1 ≤B < A ≤1, and g ∈Ap then if the condition |α| ≥ (A −B) (1 + sinh (1))2 1 + cos (1) −sin (1) −B (1 + cosh (1) + sinh (1)), and 1 + αg (z) zg′ (z)  2 + zg′′ (z) g′ (z) −zg′ (z) g (z)  ≺1 + Az 1 + Bz , holds then g ∈Gsh.

Theorem 12. For −1 ≤B < A ≤1 and f ∈Mp then if the condition (4.11) |α| ≥ (A −B) (1 + sinh (1))3 1 + cos (1) −sin (1) −B (1 + cosh (1) + sinh (1)), holds and (4.12) 1 + αz3f ′ (z) (f (z))3 ≺1 + Az 1 + Bz , then f (z) z ≺1 + sinh w (z) .

Lemma 6 for m ≥1, we have, z0w′ (z0) = mw (z0) . So we have

Corollary 5. For −1 ≤B < A ≤1, and g ∈Ap, then if the condition |α| ≥ (A −B) (1 + sinh (1))3 1 + cos (1) −sin (1) −B (1 + cosh (1) + sinh (1)) and 1 + α g (z)2 z2 (g′ (z))2  2 + zg′′ (z) g′ (z) −zg′ (z) g (z)  ≺1 + Az 1 + Bz ,