On extensions of the Open Door Lemma
-
Mostafa Amani
Abstract
The Open Door Lemma is the well-known famous result due to Miller and Mocanu that provides a sufficient condition for an analytic function on the unit disk to have positive real part. This lemma has already been corrected by Kuroki and Owa for non-real initial point. Also, Li and Sugawa extended the Open Door Lemma in a such way that an analytic function takes its values in the given symmetric sector. Now, by the same methods, we provide a sufficient condition for an analytic function to take its values in the asymmetric arbitrary sector.
The fourth author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861)
-
Communicated by Stanisława Kanas
References
[1] Duren, P. L.: Univalent Functions, Springer-Verlag, 1983.Search in Google Scholar
[2] Kuroki, K.—Owa, S.: Notes on the open door lemma, Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), 423–434.Search in Google Scholar
[3] Li, M.—Sugawa, T.: Some extensions of the Open Door Lemma, Studia Univ. Babes-Bolyai Math. 60(3) (2015), 421–430.Search in Google Scholar
[4] Miller, S. S.—Mocanu, P. T.: Briot-Bouquet differential equations and differential subordinations, Complex Variables Theory Appl. 33 (1997), 217–237.10.1080/17476939708815024Search in Google Scholar
[5] Miller, S. S.— Mocanu, P. T.: Differential Subordinations, Theory and Applications, Marcel Dekker, Inc. New York, 2000.10.1201/9781482289817Search in Google Scholar
[6] Mocanu, P. T.: On strongly-starlike and strongly-convex functions, Studia Univ. Babes-Bolyai Math. 31 (1986), 16–21.Search in Google Scholar
[7] Mocanu, P. T.: Alpha-convex integral operator and strongly starlike functions, Studia Univ. Babes-Bolyai Math. 34 (1989), 18–24.Search in Google Scholar
[8] Marx, A.:Unteruchungen Über schlicht Abbildungen, Math. Ann. 107 (1932/33), 40–67.10.1007/BF01448878Search in Google Scholar
[9] Strohhäcker, E.: Beiträge zur Theorie der schlicten Funktionen, Math. Z. 37 (1933), 356–380.10.1007/BF01474580Search in Google Scholar
© 2022 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- Logical and algebraic properties of generalized orthomodular posets
- Integral prefilters and integral Eq-algebras
- Modules with fusion and implication based over distributive lattices: Representation and duality
- Localization of k × j-rough Heyting algebras
- Meet infinite distributivity for congruence lattices of direct sums of algebras
- Some exponential diophantine equations II: The equation x2 – Dy2 = kz for even k
- On extensions of the Open Door Lemma
- Nevanlinna theory for holomophic curves from annuli into semi-Abelian varieties
- Stability and feedback stabilizability of delay periodic differential equations with pairwise permutable matrix functions
- On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions
- Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations
- Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomials
- A Study on statistical versions of convergence of sequences of functions
- Fuzzy fixed point theorems and Ulam-Hyers stability of fuzzy set-valued maps
- Notes on affine Killing and two-Killing vector fields
- Prediction of the exponential fractional upper record-values
- On concomitants of generalized order statistics from generalized FGM family under a general setting
- One-parameter generalization of dual-hyperbolic Pell numbers
Articles in the same Issue
- Regular Papers
- Logical and algebraic properties of generalized orthomodular posets
- Integral prefilters and integral Eq-algebras
- Modules with fusion and implication based over distributive lattices: Representation and duality
- Localization of k × j-rough Heyting algebras
- Meet infinite distributivity for congruence lattices of direct sums of algebras
- Some exponential diophantine equations II: The equation x2 – Dy2 = kz for even k
- On extensions of the Open Door Lemma
- Nevanlinna theory for holomophic curves from annuli into semi-Abelian varieties
- Stability and feedback stabilizability of delay periodic differential equations with pairwise permutable matrix functions
- On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions
- Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations
- Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomials
- A Study on statistical versions of convergence of sequences of functions
- Fuzzy fixed point theorems and Ulam-Hyers stability of fuzzy set-valued maps
- Notes on affine Killing and two-Killing vector fields
- Prediction of the exponential fractional upper record-values
- On concomitants of generalized order statistics from generalized FGM family under a general setting
- One-parameter generalization of dual-hyperbolic Pell numbers
