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Directional convexity of combinations of harmonic half-plane and strip mappings
Subzar Beig,Vaithiyanathan Ravichandran 대한수학회 2022 대한수학회논문집 Vol.37 No.1
For $k=1,2$, let $f_k=h_k+\overline{g_k}$ be normalized harmonic right half-plane or vertical strip mappings. We consider the convex combination $\hat{f}=\eta f_1+(1-\eta)f_2 =\eta h_1+(1-\eta)h_2 +\overline{\overline{\eta} g_1+(1-\overline{\eta})g_2}$ and the combination $\tilde{f}=\eta h_1+(1-\eta)h_2+\overline{\eta g_1+(1-\eta)g_2}$. For real $\eta$, the two mappings $\hat{f}$ and $\tilde{f}$ are the same. We investigate the univalence and directional convexity of $\hat{f}$ and $\tilde{f}$ for $\eta\in\mathbb{C}$. Some sufficient conditions are found for convexity of the combination $\tilde{f}$.
ON FUNCTIONS STARLIKE WITH RESPECT TO n-PLY SYMMETRIC, CONJUGATE AND SYMMETRIC CONJUGATE POINTS
Malik, Somya,Ravichandran, Vaithiyanathan Korean Mathematical Society 2022 대한수학회논문집 Vol.37 No.4
For given non-negative real numbers 𝛼<sub>k</sub> with ∑<sup>m</sup><sub>k=1</sub> 𝛼<sub>k</sub> = 1 and normalized analytic functions f<sub>k</sub>, k = 1, …, m, defined on the open unit disc, let the functions F and Fn be defined by F(z) := ∑<sup>m</sup><sub>k=1</sub> 𝛼<sub>k</sub>f<sub>k</sub>(z), and F<sub>n</sub>(z) := n<sup>-1</sup> ∑<sup>n-1</sup><sub>j=0</sub> e<sup>-2j𝜋i/n</sup>F(e<sup>2j𝜋i/n</sup>z). This paper studies the functions f<sub>k</sub> satisfying the subordination zf'<sub>k</sub>(z)/F<sub>n</sub>(z) ≺ h(z), where the function h is a convex univalent function with positive real part. We also consider the analogues of the classes of starlike functions with respect to symmetric, conjugate, and symmetric conjugate points. Inclusion and convolution results are proved for these and related classes. Our classes generalize several well-known classes and the connections with the previous works are indicated.
Sufficient conditions for starlikeness of reciprocal order
Saravanarasu Madhumitha,Vaithiyanathan Ravichandran 강원경기수학회 2023 한국수학논문집 Vol.31 No.3
A normalized analytic function $f$ defined on the unit disk $\mathbb{D}$ is starlike of reciprocal order $\alpha$, $0\leq \alpha<1$, if $\operatorname{Re}(f(z)/(zf'(z)))>\alpha$ for all $z\in \mathbb{D}$. Such functions are starlike and therefore univalent in $\mathbb{D}$. Using the well-known Miller-Mocanu differential subordination theory, sufficient conditions involving differential inclusions are obtained for a normalized analytic function to be starlike of reciprocal order $\alpha$. Furthermore, a few conditions are derived for a function $f$ to belong to a subclass of reciprocal starlike functions, satisfying $\left\lvert f(z)/ (z f'(z)) - 1 \right\rvert < 1-\alpha$.
INCLUSION RELATIONS AND RADIUS PROBLEMS FOR A SUBCLASS OF STARLIKE FUNCTIONS
Gupta, Prachi,Nagpal, Sumit,Ravichandran, Vaithiyanathan Korean Mathematical Society 2021 대한수학회지 Vol.58 No.5
By considering the polynomial function 𝜙<sub>car</sub>(z) = 1 + z + z<sup>2</sup>/2, we define the class 𝓢<sup>*</sup><sub>car</sub> consisting of normalized analytic functions f such that zf'/f is subordinate to 𝜙<sub>car</sub> in the unit disk. The inclusion relations and various radii constants associated with the class 𝓢<sup>*</sup><sub>car</sub> and its connection with several well-known subclasses of starlike functions is established. As an application, the obtained results are applied to derive the properties of the partial sums and convolution.
FUNCTIONS SUBORDINATE TO THE EXPONENTIAL FUNCTION
Priya G. Krishnan,Vaithiyanathan Ravichandran,Ponnaiah Saikrishnan Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.1
We use the theory of differential subordination to explore various inequalities that are satisfied by an analytic function p defined on the unit disc so that the function p is subordinate to the function e<sup>z</sup>. These results are applied to find sufficient conditions for the normalised analytic functions f defined on the unit disc to satisfy the subordination zf'(z)/f(z) ≺ e<sup>z</sup>.
THE THIRD HERMITIAN-TOEPLITZ AND HANKEL DETERMINANTS FOR PARABOLIC STARLIKE FUNCTIONS
Rosihan M. Ali,Sushil Kumar,Vaithiyanathan Ravichandran 대한수학회 2023 대한수학회보 Vol.60 No.2
A normalized analytic function $f$ is parabolic starlike if $w(z)$ $:=zf'(z)/f(z)$ maps the unit disk into the parabolic region $\{w: \operatorname{Re} w>|w-1|\}$. Sharp estimates on the third Hermitian-Toeplitz determinant are obtained for parabolic starlike functions. In addition, upper bounds on the third Hankel determinants are also determined.