scholarly journals Third Hankel determinant for reciprocal of bounded turning function has a positive real part of order alpha

2017 ◽  
Vol 62 (3) ◽  
pp. 331-340
Author(s):  
Bolineni Venkateswarlu ◽  
◽  
Nekkanti Rani ◽  
Keyword(s):  
Positive Real Part ◽  
Hankel Determinant ◽  
Positive Real ◽  
Bounded Turning

scholarly journals On Coefficient Functionals for Functions with Coefficients Bounded by 1

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 491
Author(s):  
Paweł Zaprawa ◽  
Anna Futa ◽  
Magdalena Jastrzębska
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Extremal Functions ◽  
Hankel Determinant ◽  
Schwarz Function ◽  
Positive Real ◽  
Special Case

In this paper, we discuss two well-known coefficient functionals a 2 a 4 - a 3 2 and a 4 - a 2 a 3 . The first one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class Q R ( 1 2 ) of analytic functions with real coefficients which satisfy the condition ( ) f ( z ) z > 1 2 for z in the unit disk Δ . It is known that all coefficients of f ∈ Q R ( 1 2 ) are bounded by 1. We find the upper bound of a 2 a 4 - a 3 2 and the bound of | a 4 - a 2 a 3 | . We also consider a few subclasses of Q R ( 1 2 ) and we estimate the above mentioned functionals. In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates significantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefficient functionals considered in the case of real coefficients and arbitrary coefficients.

Second hankel determinat for certain analytic functions satisfying subordinate condition

2018 ◽  
Vol 68 (2) ◽  
pp. 463-471
Author(s):  
Erhan Deniz ◽  
Levent Budak
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Real Axis ◽  
Upper Bound ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Hankel Determinant ◽  
Positive Real ◽  
Toeplitz Determinants ◽  
Second Hankel Determinant

Abstract In this paper, we introduce and investigate the following subclass $$\begin{array}{} \displaystyle 1+\frac{1}{\gamma }\left( \frac{zf'(z)+\lambda z^{2}f''(z)}{\lambda zf'(z)+(1-\lambda )f(z)}-1\right) \prec \varphi (z)\qquad\left( 0\leq \lambda \leq 1,\gamma \in \mathbb{C} \smallsetminus \{0\}\right) \end{array} $$ of analytic functions, φ is an analytic function with positive real part in the unit disk 𝔻, satisfying φ (0) = 1, φ '(0) > 0, and φ (𝔻) is symmetric with respect to the real axis. We obtain the upper bound of the second Hankel determinant | a2a4− $\begin{array}{} a^2_3 \end{array} $ | for functions belonging to the this class is studied using Toeplitz determinants. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

scholarly journals Initial logarithmic coefficients for functions starlike with respect to symmetric points

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Paweł Zaprawa
Keyword(s):  
Exact Result ◽  
Positive Real Part ◽  
Additional Benefit ◽  
Extremal Functions ◽  
Positive Real ◽  
Logarithmic Coefficients ◽  
Symmetric Points

AbstractIn this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S .

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