# Some subclasses of close-to-convex functions

• Volume: 58, Issue: 1, page 53-64
• ISSN: 0066-2216

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## Abstract

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For α ∈ [0,1] and β ∈ (-π/2,π/2) we introduce the classes ${C}_{\beta }\left(\alpha \right)$ defined as follows: a function f regular in U = z: |z| < 1 of the form $f\left(z\right)=z+{\sum }_{n=1}^{\infty }{a}_{n}{z}^{n}$, z ∈ U, belongs to the class ${C}_{\beta }\left(\alpha \right)$ if $Re{e}^{i\beta }\left(1-\alpha ²z²\right){f}^{\text{'}}\left(z\right)<0$ for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in ${C}_{\beta }\left(\alpha \right)$ are examined.

## How to cite

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Adam Lecko. "Some subclasses of close-to-convex functions." Annales Polonici Mathematici 58.1 (1993): 53-64. <http://eudml.org/doc/262387>.

abstract = {For α ∈ [0,1] and β ∈ (-π/2,π/2) we introduce the classes $C_β(α)$ defined as follows: a function f regular in U = z: |z| < 1 of the form $f(z) = z + ∑_\{n=1\}^\{∞\} a_n z^n$, z ∈ U, belongs to the class $C_β(α)$ if $Re\{e^\{iβ\}(1 - α²z²)f^\{\prime \}(z)\} < 0$ for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in $C_β(α)$ are examined.},
author = {Adam Lecko},
journal = {Annales Polonici Mathematici},
keywords = {close-to-convex functions; close-to-convex functions with argument β; functions convex in the direction of the imaginary axis; functions of bounded rotation with argument β; distortion theorems},
language = {eng},
number = {1},
pages = {53-64},
title = {Some subclasses of close-to-convex functions},
url = {http://eudml.org/doc/262387},
volume = {58},
year = {1993},
}

TY - JOUR
AU - Adam Lecko
TI - Some subclasses of close-to-convex functions
JO - Annales Polonici Mathematici
PY - 1993
VL - 58
IS - 1
SP - 53
EP - 64
AB - For α ∈ [0,1] and β ∈ (-π/2,π/2) we introduce the classes $C_β(α)$ defined as follows: a function f regular in U = z: |z| < 1 of the form $f(z) = z + ∑_{n=1}^{∞} a_n z^n$, z ∈ U, belongs to the class $C_β(α)$ if $Re{e^{iβ}(1 - α²z²)f^{\prime }(z)} < 0$ for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in $C_β(α)$ are examined.
LA - eng
KW - close-to-convex functions; close-to-convex functions with argument β; functions convex in the direction of the imaginary axis; functions of bounded rotation with argument β; distortion theorems
UR - http://eudml.org/doc/262387
ER -

## References

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7. [7] T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532-537. Zbl0106.04805
8. [8] P. T. Mocanu, Une propriété de convexité généralisée dans la théorie de la représentation conforme, Mathematica (Cluj) 11 (34) (1969), 127-133. Zbl0195.36401
9. [9] K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. Jap. (1) 2 (1934-1935), 129-155. Zbl0010.26305
10. [10] L. Špaček, Contribution à la theorie des fonctions univalentes, Časopis Pěst. Mat. 2 (1932), 12-19. Zbl58.0365.04
11. [11] J. Stankiewicz and J. Waniurski, Some classes of functions subordinate to linear transformation and their applications, Ann. Univ. Mariae Curie-Skłodowska Sect. A 28 (1974), 85-94.
12. [12] S. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38 (1935), 310-340. Zbl0014.26707

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