Ma-Minda φ-classes studied in this paper:
Abstract
In this article, we find a condition on α so that if 1 + αz z f ′(z)
f(z) ≺1+Az
1+Bz, then zF ′(z)
F (z) ≺
1+sin(z), where
F (z) = γ+1
zβ
Z z
0 tγ−1 f (t)dt,
is the well-known Bernardi integral operator. We also study the case zF ′(z)
F (z) ≺cos(z). Some
similar implications are also discussed for both functions.
2010 Mathematics Subject Classification: 30C45; 30C50
Results & Lemmas (19)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1
Lemma 1 ([8]). Let w be a non-constant analytic function in D with w(0) = 0. If |w(z0)| = max |w(z)|, |z| ≤|z0|, z ∈D, then there exists a…
Lemma 1 ([8]). Let w be a non-constant analytic function in D with w(0) = 0. If |w(z0)| = max{|w(z)|, |z| ≤|z0|}, z ∈D, then there exists a real number m (m ≥1) such that z0w′ (z0) = mw(z0). 2. BERNARDI’S INTEGRAL OPERATOR ASSOCIATED WITH SINE FUNCTION
Theorem 1.
Theorem 1. Assume that |α| ≥ (A−B)(1+γ+sinh(1)) (1−|B|)cos(1)−(1+|B|)(1+γ+sinh(1))(1+sinh(1)). (2.1) If 1+αz z f ′ (z) f (z) ≺1+Az 1+Bz,…
Theorem 1. Assume that |α| ≥ (A−B)(1+γ+sinh(1)) (1−|B|)cos(1)−(1+|B|)(1+γ+sinh(1))(1+sinh(1)). (2.1) If 1+αz z f ′ (z) f (z) ≺1+Az 1+Bz, −1 ≤B < A ≤1, (2.2) then zF ′ (z)
Corollary 1.
Corollary 1. Let |α| ≥ (1+γ+sinh(1)) cos(1)−(1+γ+sinh(1))(1+sinh(1)). If 1+αz zf ′ (z) f (z) ≺1+z, then zF ′ (z) F (z) ≺1+sin(z). By…
Corollary 1. Let |α| ≥ (1+γ+sinh(1)) cos(1)−(1+γ+sinh(1))(1+sinh(1)). If 1+αz zf ′ (z) f (z) ≺1+z, then zF ′ (z) F (z) ≺1+sin(z). By choosing A = 1, B = −1, we have the following result.
Corollary 2.
Corollary 2. Let |α| ≥−0.363370517 and 1+αz zf ′ (z) f (z) ≺1+z 1−z. Then zF ′ (z) F (z) ≺1+sin(z).
Corollary 2. Let |α| ≥−0.363370517 and 1+αz zf ′ (z) f (z) ≺1+z 1−z. Then zF ′ (z) F (z) ≺1+sin(z).
Theorem 2.
Theorem 2. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)cos(1)−(1+|B|)(1+γ)(1+sinh(1)). (2.7)
Theorem 2. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)cos(1)−(1+|B|)(1+γ)(1+sinh(1)). (2.7)
Corollary 3.
Corollary 3. Let |α| ≥ (γ+1) cos(1)−(1+|B|)(1+γ)(1+sinh(1)) and 1+αf (z) ≺1+z. Then F (z) z ≺1+sin(z). By choosing A = 1, B = −1, we have…
Corollary 3. Let |α| ≥ (γ+1) cos(1)−(1+|B|)(1+γ)(1+sinh(1)) and 1+αf (z) ≺1+z. Then F (z) z ≺1+sin(z). By choosing A = 1, B = −1, we have the following result.
Corollary 4.
Corollary 4. Let |α| ≥ −1 1+sinh(1) and 1+αf (z) ≺1+z 1−z. Then F (z) z ≺1+sin(z).
Corollary 4. Let |α| ≥ −1 1+sinh(1) and 1+αf (z) ≺1+z 1−z. Then F (z) z ≺1+sin(z).
Theorem 3.
Theorem 3. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)cos(1)−(1+|B|)(1+γ)(1+sinh(1)). (2.12) If 1+αzf ′ (z) ≺1+Az 1+Bz, (2.13) then F ′ (z)…
Theorem 3. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)cos(1)−(1+|B|)(1+γ)(1+sinh(1)). (2.12) If 1+αzf ′ (z) ≺1+Az 1+Bz, (2.13) then F ′ (z) ≺1+sin(z),
Corollary 5.
Corollary 5. Assume that |α| ≥ (γ+1) cos(1)−(1+γ)(1+sinh(1)). If 1+αzf ′ (z) ≺1+z, then F ′ (z) ≺1+sin(z). By choosing A = 1, B = −1, we…
Corollary 5. Assume that |α| ≥ (γ+1) cos(1)−(1+γ)(1+sinh(1)). If 1+αzf ′ (z) ≺1+z, then F ′ (z) ≺1+sin(z). By choosing A = 1, B = −1, we have the following result.
Corollary 6.
Corollary 6. Assume that |α| ≥ −1 (1+sinh(1)) and if 1+αzf ′ (z) ≺1+z 1−z, then F ′ (z) ≺1+sin(z). 3. BERNARDI’S INTEGRAL OPERATOR…
Corollary 6. Assume that |α| ≥ −1 (1+sinh(1)) and if 1+αzf ′ (z) ≺1+z 1−z, then F ′ (z) ≺1+sin(z). 3. BERNARDI’S INTEGRAL OPERATOR ASSOCIATED WITH COSINE FUNCTION
Theorem 4.
Theorem 4. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)sin(1)−(1+|B|)(cosh(1)+γ)cosh(1). (3.1) If 1+αz zf ′ (z) f (z) ≺1+Az 1+Bz, (3.2) then zF…
Theorem 4. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)sin(1)−(1+|B|)(cosh(1)+γ)cosh(1). (3.1) If 1+αz zf ′ (z) f (z) ≺1+Az 1+Bz, (3.2) then zF ′ (z)
Corollary 7.
Corollary 7. Assume that |α| ≥ (γ+1) sin(1)−(cosh(1)+γ)cosh(1). If 1+αz zf ′ (z) f (z) ≺1+z, then zF ′ (z) F (z) ≺cos(z),
Corollary 7. Assume that |α| ≥ (γ+1) sin(1)−(cosh(1)+γ)cosh(1). If 1+αz zf ′ (z) f (z) ≺1+z, then zF ′ (z) F (z) ≺cos(z),
Corollary 8.
Corollary 8. Assume that |α| ≥ −(γ+1) (cosh(1)+γ)cosh(1). If 1+αz zf ′ (z) f (z) ≺1+z 1−z, then zF ′ (z) F (z) ≺cos(z).
Corollary 8. Assume that |α| ≥ −(γ+1) (cosh(1)+γ)cosh(1). If 1+αz zf ′ (z) f (z) ≺1+z 1−z, then zF ′ (z) F (z) ≺cos(z).
Theorem 5.
Theorem 5. Assume that α ≥ (A−B)(γ+1) (1−|B|)sin(1)−(1+|B|)(cosh(1)+γ)cosh(1). (3.4) If 1+αf (z) ≺1+Az 1+Bz, (3.5) then F (z) z ≺cos(z),…
Theorem 5. Assume that α ≥ (A−B)(γ+1) (1−|B|)sin(1)−(1+|B|)(cosh(1)+γ)cosh(1). (3.4) If 1+αf (z) ≺1+Az 1+Bz, (3.5) then F (z) z ≺cos(z), where F is the Bernardi integral operator defined in (1.3).
Corollary 9.
Corollary 9. Assume that |α| ≥ (γ+1) sin(1)−(cosh(1)+γ)cosh(1). If 1+αf (z) ≺1+z, then F (z) z ≺cos(z). By choosing A = 1, B = −1, we have…
Corollary 9. Assume that |α| ≥ (γ+1) sin(1)−(cosh(1)+γ)cosh(1). If 1+αf (z) ≺1+z, then F (z) z ≺cos(z). By choosing A = 1, B = −1, we have the following result.
Corollary 10.
Corollary 10. Assume that |α| ≥ −(γ+1) (cosh(1)+γ)cosh(1).
Corollary 10. Assume that |α| ≥ −(γ+1) (cosh(1)+γ)cosh(1).
Theorem 6.
Theorem 6. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)sin(1)−(1+|B|)(1+γ)cosh(1). (3.9) If 1+αzf ′ (z) ≺1+Az 1+Bz, (3.10) then F ′ (z) ≺cos(z),…
Theorem 6. Assume that |α| ≥ (A−B)(γ+1) (1−|B|)sin(1)−(1+|B|)(1+γ)cosh(1). (3.9) If 1+αzf ′ (z) ≺1+Az 1+Bz, (3.10) then F ′ (z) ≺cos(z), where F is the Bernardi integral operator defined in (1.3).
Corollary 11.
Corollary 11. Assume that |α| ≥ (γ+1) sin(1)−(1+γ)cosh(1). If 1+αzf ′ (z) ≺1+z, then F ′ (z) ≺cos(z). By choosing A = 1, B = −1, we have…
Corollary 11. Assume that |α| ≥ (γ+1) sin(1)−(1+γ)cosh(1). If 1+αzf ′ (z) ≺1+z, then F ′ (z) ≺cos(z). By choosing A = 1, B = −1, we have the following result.
Corollary 12.
Corollary 12. Assume that |α| ≥ −(γ+1) (1+γ)cosh(1). If 1+αzf ′ (z) ≺1+z 1−z, then F ′ (z) ≺cos(z).
Corollary 12. Assume that |α| ≥ −(γ+1) (1+γ)cosh(1). If 1+αzf ′ (z) ≺1+z 1−z, then F ′ (z) ≺cos(z).
Function classes studied:
Related Papers