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MONOTONICITY PROPERTIES OF THE GENERALIZED STRUVE FUNCTIONS
Rosihan M. Ali,Saiful R. Mondal,Kottakkaran S. Nisar 대한수학회 2017 대한수학회지 Vol.54 No.2
This paper introduces and studies a generalization of the classical Struve function of order $p$ given by \[{}_a\mathtt{S}_{p, c}(x):= \sum_{k=0}^\infty \frac{(-c)^k}{\mathrm{\Gamma}{\left( a k +p+\frac{3}{2}\right)} \mathrm{\Gamma}{\left( k+\frac{3}{2}\right)}} \left(\frac{x}{2}\right)^{2k+p+1}.\] Representation formulae are derived for ${}_a\mathtt{S}_{p, c}.$ Further the function ${}_a\mathtt{S}_{p, c}$ is shown to be a solution of an $(a+1)$-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function ${}_a\mathtt{S}_{p, c}$ are investigated, particulary for the case $c=-1$. As a consequence, Tur\'an-type inequalities are established. For $a=2$ and $c=-1,$ dominant and subordinant functions are obtained for the Struve function ${}_2\mathtt{S}_{p, -1}.$
Ali, Rosihan M.,Chandrashekar, R.,Lee, See-Keong,Swaminathan, A.,Ravichandran, V. Department of Mathematics 2011 Kyungpook mathematical journal Vol.51 No.2
Differential subordination and superordination results are obtained for multivalent meromorphic functions associated with the Liu-Srivastava linear operator in the punctured unit disk. These results are derived by investigating appropriate classes of admissible functions. Sandwich-type results are also obtained.
MONOTONICITY PROPERTIES OF THE GENERALIZED STRUVE FUNCTIONS
Ali, Rosihan M.,Mondal, Saiful R.,Nisar, Kottakkaran S. Korean Mathematical Society 2017 대한수학회지 Vol.54 No.2
This paper introduces and studies a generalization of the classical Struve function of order p given by $$_aS_{p,c}(x):=\sum\limits_{k=0}^{\infty}\frac{(-c)^k}{{\Gamma}(ak+p+\frac{3}{2}){\Gamma}(k+\frac{3}{2})}(\frac{x}{2})^{2k+p+1}$$. Representation formulae are derived for $_aS_{p,c}$. Further the function $_aS_{p,c}$ is shown to be a solution of an (a + 1)-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function $_aS_{p,c}$ are investigated, particulary for the case c = -1. As a consequence, $Tur{\acute{a}}n$-type inequalities are established. For a = 2 and c = -1, dominant and subordinant functions are obtained for the Struve function $_2S_{p,-1}$.
Radius of Starlikeness for Analytic Functions with Fixed Second Coefficient
Ali, Rosihan M.,Kumar, Virendra,Ravichandran, V.,Kumar, Shanmugam Sivaprasad Department of Mathematics 2017 Kyungpook mathematical journal Vol.57 No.3
Sharp radius constants for certain classes of normalized analytic functions with fixed second coefficient, to be in the classes of starlike functions of positive order, parabolic starlike functions, and Sokół-Stankiewicz starlike functions are obtained. Our results extend several earlier works.
Undergraduate Mathematics Enhanced With Graphing Technology
Ali, Rosihan M.,Kee, Kor Liew 한국수학교육학회 2004 수학교육연구 Vol.8 No.1
The school of Mathematical Science at University Sains Malaysia has offered a laboratory course on the integration of hand-held technology into the teaching and learning of mathematics since the beginning of the 2001/2002 academic year. This inquiry-based course highlights the explorations and application of mathematics in a data rich modeling environment. In addition, the course addresses several issues related to the effective integration of such technology into the mathematics curriculum. This paper discusses the appropriate use of graphing technology to present mathematical concepts and to support student's understanding in a student-centered learning environment, shares knowledge on the new mathematics that was made possible by hand-held technology, and summarizes student reactions to this innovative learning mode.
Sharp bounds for initial coefficients and the second Hankel determinant
Rosihan M. Ali,See Keong Lee,Milutin Obradovic 대한수학회 2020 대한수학회보 Vol.57 No.4
For functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ belonging to particular classes, this paper finds sharp bounds for the initial coefficients $a_{2}, \, a_{3}, \, a_{4},$ as well as the sharp estimate for the second order Hankel determinant $H_{2}(2)=a_{2}a_{4}-a_{3}^{2}.$ Two classes are treated: first is the class consisting of $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in the unit disk $\mathbb{D}$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda, \quad 0<\alpha <1, \, 0 < \lambda \leq 1.$$ The second class consists of Bazilevi\v{c} functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in $\mathbb{D}$ satisfying $${\rm Re}\left\{\left(\frac{f(z)}{z}\right)^{\alpha-1}f'(z)\right\}>0, \quad \alpha >0.$$
SHARP BOUNDS FOR INITIAL COEFFICIENTS AND THE SECOND HANKEL DETERMINANT
Ali, Rosihan M.,Lee, See Keong,Obradovic, Milutin Korean Mathematical Society 2020 대한수학회보 Vol.57 No.4
For functions f(z) = z + a<sub>2</sub>z<sup>2</sup> + a<sub>3</sub>z<sup>3</sup> + ⋯ belonging to particular classes, this paper finds sharp bounds for the initial coefficients a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, as well as the sharp estimate for the second order Hankel determinant H<sub>2</sub>(2) = a<sub>2</sub>a<sub>4</sub> - a<sub>2</sub><sup>3</sup>. Two classes are treated: first is the class consisting of f(z) = z + a<sub>2</sub>z<sup>2</sup> + a<sub>3</sub>z<sup>3</sup> + ⋯ in the unit disk 𝔻 satisfying $$\|\(\frac{z}{f(z)}\)^{1+{\alpha}}\;f^{\prime}(z)-1\|<{\lambda},\;0<{\alpha}<1,\;0<{\lambda}{\leq}1.$$ The second class consists of Bazilevič functions f(z) = z+a<sub>2</sub>z<sup>2</sup>+a<sub>3</sub>z<sup>3</sup>+⋯ in 𝔻 satisfying $$Re\{\(\frac{f(z)}{z}\)^{{\alpha}-1}\;f^{\prime}(z)\}>0,\;{\alpha}>0.$$
A CLASS OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS DEFINED BY CONVOLUTION
Ali Rosihan M.,Khan M. Hussain,Ravichandran V.,Subramanian K.G. Korean Mathematical Society 2006 대한수학회보 Vol.43 No.1
For a given p-valent analytic function g with positive coefficients in the open unit disk $\Delta$, we study a class of functions $f(z) = z^p - \sum\limits{_{n=m}}{^\infty} a_nz^n(a_n{\geq}0)$ satisfying $$\frac 1 {p}{\Re}\;(\frac {z(f*g)'(z)} {(f*g)(z)})\;>\;\alpha\;(0{\leq}\;\alpha\;<\;1;z{\in}{\Delta})$$ Coefficient inequalities, distortion and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases.