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On Convex Functions with Complex Order Through Bounded Boundary Rotation

On Convex Functions with Complex Order Through Bounded Boundary Rotation Let F be the class of functions $$f(z)=z+a_{2}z^{2}+\cdots $$ f ( z ) = z + a 2 z 2 + ⋯ which are analytic in $${\mathcal {D}}=\{z: |z|<1\}$$ D = { z : | z | < 1 } and satisfies the condition $$\begin{aligned} 1+\frac{1}{b}z\frac{f''(z)}{f'(z)}=p_{t}(z), (b\ne 0, b\in {\mathcal {C}}, z\in {\mathcal {D}}) \end{aligned}$$ 1 + 1 b z f ′ ′ ( z ) f ′ ( z ) = p t ( z ) , ( b ≠ 0 , b ∈ C , z ∈ D ) where $$p_{t}(z)=\left( \frac{t}{4}+\frac{1}{2}\right) p_{1}(z)-\left( \frac{t}{4}-\frac{1}{2}\right) p_{2}(z)$$ p t ( z ) = t 4 + 1 2 p 1 ( z ) - t 4 - 1 2 p 2 ( z ) , $$t\ge 2, p_{1}(z),p_{2}(z)\in {\mathcal {P}}$$ t ≥ 2 , p 1 ( z ) , p 2 ( z ) ∈ P . $${\mathcal {P}}$$ P is the class of analytic functions with the positive real part (Caratheodory class) then this function will be called convex function by means of bounded boundary rotation and denoted by K(t, b). In this present paper, we will introduce this class and its some properties. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics in Computer Science Springer Journals

On Convex Functions with Complex Order Through Bounded Boundary Rotation

Mathematics in Computer Science , Volume 13 (3) – Jul 4, 2019

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Mathematics, general; Computer Science, general
ISSN
1661-8270
eISSN
1661-8289
DOI
10.1007/s11786-019-00405-8
Publisher site
See Article on Publisher Site

Abstract

Let F be the class of functions $$f(z)=z+a_{2}z^{2}+\cdots $$ f ( z ) = z + a 2 z 2 + ⋯ which are analytic in $${\mathcal {D}}=\{z: |z|<1\}$$ D = { z : | z | < 1 } and satisfies the condition $$\begin{aligned} 1+\frac{1}{b}z\frac{f''(z)}{f'(z)}=p_{t}(z), (b\ne 0, b\in {\mathcal {C}}, z\in {\mathcal {D}}) \end{aligned}$$ 1 + 1 b z f ′ ′ ( z ) f ′ ( z ) = p t ( z ) , ( b ≠ 0 , b ∈ C , z ∈ D ) where $$p_{t}(z)=\left( \frac{t}{4}+\frac{1}{2}\right) p_{1}(z)-\left( \frac{t}{4}-\frac{1}{2}\right) p_{2}(z)$$ p t ( z ) = t 4 + 1 2 p 1 ( z ) - t 4 - 1 2 p 2 ( z ) , $$t\ge 2, p_{1}(z),p_{2}(z)\in {\mathcal {P}}$$ t ≥ 2 , p 1 ( z ) , p 2 ( z ) ∈ P . $${\mathcal {P}}$$ P is the class of analytic functions with the positive real part (Caratheodory class) then this function will be called convex function by means of bounded boundary rotation and denoted by K(t, b). In this present paper, we will introduce this class and its some properties.

Journal

Mathematics in Computer ScienceSpringer Journals

Published: Jul 4, 2019

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