Some subclasses of close-to-convex functions

Adam Lecko

Annales Polonici Mathematici (1993)

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

Abstract

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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 R e e i β ( 1 - α ² z ² ) f ' ( z ) < 0 for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in C β ( α ) 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>.

@article{AdamLecko1993,
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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  3. [3] A. W. Goodman and E. B. Saff, On the definition of a close-to-convex function, Internat. J. Math. and Math. Sci. 1 (1978), 125-132. Zbl0383.30005
  4. [4] W. Hengartner and G. Schober, On schlicht mappings to domains convex in one direction, Comment. Math. Helv. 45 (1970), 303-314. Zbl0203.07604
  5. [5] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185. Zbl0048.31101
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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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