scholarly journals On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

2021 ◽  
Vol 27 (1) ◽  
pp. 93-106
Author(s):  
Toshiyuki SUGAWA ◽  
Li-Mei WANG
Keyword(s):  
Starlike Function ◽  
Positive Order

scholarly journals On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

2020 ◽  
Author(s):  
Toshiyuki Sugawa ◽  
Li-Mei Wang
Keyword(s):  
Extremal Problem ◽  
Unit Disk ◽  
Complex Plane ◽  
Inverse Function ◽  
Starlike Function ◽  
Extremal Functions ◽  
Positive Order ◽  
Sharp Estimates ◽  
The Third ◽  
Straightforward Proof

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0<\alpha<1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

scholarly journals Growth results for a subclass of Bazilevič functions

1985 ◽  
Vol 8 (4) ◽  
pp. 785-793
Author(s):  
Rabha Md. El-Ashwah ◽  
D. K. Thomas
Keyword(s):  
Unit Disc ◽  
Starlike Function ◽  
Area Function ◽  
Proper Subclass ◽  
Bazilevic Functions

Forα>0, letB(α)be the class of regular normalized Bazilevič functions defined in the unit disc. Choosing the associated starlike functiong(z)≡zgives a proper subclassB1(α)ofB(α). ForB(α), correct growth estimates in terms of the area function are unknown. Several results in this direction are given forB1(12).

scholarly journals On Absolute Summability for any Positive Order

1933 ◽  
Vol 3 (3) ◽  
pp. 173-178 ◽  
Author(s):  
C. E. Winn
Keyword(s):  
Absolute Summability ◽  
Positive Order ◽  
Partial Sum ◽  
Image Position

Absolute summability according to Cesàro's method has been defined by Fekete for positive integral orders, as follows:—Denoting the rth partial sum of a series Σun by and its rth mean, namely2, by we can regard as the sum of the series.

scholarly journals On Bazilevic functions

1987 ◽  
Vol 10 (1) ◽  
pp. 79-88 ◽  
Author(s):  
Khalida I. Noor ◽  
Sumayya A. Al-Bany
Keyword(s):  
Unit Disc ◽  
Starlike Function ◽  
Hankel Determinant ◽  
Bazilevic Functions

LetB(β)be the class of Bazilevic functions of typeβ(β>0). A functionf ϵ B(β)if it is analytic in the unit discEandRezf′(z)f1−β(z)gβ(z)>0, wheregis a starlike function. We generalize the classB(β)by takinggto be a function of radius rotation at mostkπ(k≥2). Archlength, difference of coefficient, Hankel determinant and some other problems are solved for this generalized class. Fork=2, we obtain some of these results for the classB(β)of Bazilevic functions of typeβ.

On the Points of Inflection of Bessel Functions of Positive Order, I

1990 ◽  
Vol 42 (5) ◽  
pp. 933-948 ◽  
Author(s):  
Lee Lorch ◽  
Peter Szego
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Primary Concern ◽  
Second Derivative ◽  
Positive Order ◽  
Fixed Rank

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

On weak convergence within the hnbue family of life distributions

1984 ◽  
Vol 21 (03) ◽  
pp. 654-660 ◽  
Author(s):  
Sujit K. Basu ◽  
Manish C. Bhattacharjee
Keyword(s):  
Weak Convergence ◽  
Exponential Distribution ◽  
Limiting Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moment Sequence ◽  
Positive Order ◽  
Life Distributions ◽  
Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050005
Author(s):  
JIA YAO ◽  
YING CHEN ◽  
JUNQIAO LI ◽  
BIN WANG
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order ◽  
One Dimensional ◽  
Katugampola Fractional Integral ◽  
Hadamard Fractional Integral

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [Formula: see text]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

scholarly journals Convex and starlike criteria

1999 ◽  
Vol 22 (1) ◽  
pp. 75-79 ◽  
Author(s):  
Herb Silverman
Keyword(s):  
Sufficient Conditions ◽  
Starlike Functions ◽  
Positive Order ◽  
Convex And Starlike Functions

We investigate an expression involving the quotient of the analytic representations of convex and starlike functions. Sufficient conditions are found for functions to be starlike of a positive order and convex.

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