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A linear operator and associated families of meromorphically multivalent functions of order a
M. K. Aouf,H. M. Srivastava 장전수학회 2006 Advanced Studies in Contemporary Mathematics Vol.13 No.1
Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses $Q_{a,c}(p,\alpha ;A,B)$ and $Q_{a,c}^{+}(p,\alpha ;A,B)$ of meromorphically multivalent functions of order $\alpha$ $(0\leqq \alpha <p)$ in the punctured unit disk $\mathbb{U}^{\ast}$. The main object of the present paper is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions. We also derive many interesting results for the Hadamard products of functions belonging to the function class $Q_{a,c}^{+}(p,\alpha;A,B)$.
Aouf, M.K.,Shamandy, A.,Mostafa, A.O.,Adwan, Eman A. Department of Mathematics 2012 Kyungpook mathematical journal Vol.52 No.2
In this paper, we obtain some applications of first order differential subordination and superordination results involving the operator $J_{s,b}^{{\lambda},p}$ for certain normalized p-valent analytic functions associated with that operator.
ON SUBCLASSES OF P-VALENT FUNCTIONS STARLIKE IN THE UNIT DISC
Aouf, M.K. Department of Mathematics 1988 Kyungpook mathematical journal Vol.28 No.2
For a positive integer p, $A_p$ will denote the class of functions $f(z)=z^p+\sum\limits^{\infty}_{n=p+1}a_nz^n$ which are analytic in the unit disc U = {z: |z| <1}. For $0{\leq}{\alpha}{\leq}1$, 0<${\beta}{\leq}1$, $0{\leq}{\lambda}$ <p, let $S_p({\alpha},{\beta},{\lambda})$ denote the class of functions $f(z){\in}A_p$ which satisfy the condition $\left|\frac{{\frac{zf^{\prime}(z)}{f(z)}}-p}{{{\alpha}{\frac{zf^{\prime}(z)}{f(z)}}+p-{\lambda}(1+{\alpha})}}\right|$<${\beta}$ for $z{\in}U$ In this paper we obtain a representation theorem for the class $S_p({\alpha},{\beta},{\lambda})$ and also derive distortion theorem and sharp estimates for the coefficients of this class.
Convex Subclass of Starlike Functions
M. K. Aouf ...et al KYUNGPOOK UNIVERSITY 2000 Kyungpook mathematical journal Vol.40 No.2
Let T denote the class of functions of the form ◁수식삽입▷(원문을참조하세요) that are analytic and univalent in the unit disc U. In this paper we obtain a relationship between the subclasses T(λ, α) and K(λ, α) of Tby defining a subclass B(λ, α) of K(λ, α). Coefficient estimate, distortion and covering theorems are obtained for the class B(λ, α). The class B(λ, α) is convex. In terms of the operators of fractional calculus we derive several sharp results depicting the growth and distortion properties of functions belonging to the class B(λ, α).
M. K. Aouf,Tamer M. Seoudy 대한수학회 2011 대한수학회보 Vol.48 No.3
In this paper, we obtain some applications of first order differential subordination and superordination results for higher-order derivatives of p-valent functions involving certain linear operator. Some of our results improve and generalize previously known results.
ON CERTAIN SUBCLASSES OF ANALYTIC P-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
Aouf, M.K. The Youngnam Mathematical Society Korea 1989 East Asian mathematical journal Vol.5 No.1
Let $S_p*({\alpha},{\beta},{\mu})$ denote the class of functions $f(z)=z^p-{\sum}{\limit}^{\infty}_{n=1}a_{p+n}\;z^{p+n}(a_{p+n}{\geq}o,\;p{\in}N)$ analytic and p-valent in the unit disc $U=\{z:{\mid}z{\mid}<1\}$ and satisfy the condition $${\mid}\frac{\frac{zf'(z)}{f(z)}-p}{\mu\frac{zf'(z)}{f(z)}+p-(1+\mu)\alpha}\mid<\beta,\;z{\in}U$$, where $o{\leq}{\alpha}<p,\;o<{\beta}{\leq}1$ and $o\leq\mu\leq1$. Further f(z) is said to belong to the class $C_p*({\alpha},{\beta},{\mu})\;if\;zf'(z)/p{\in}S_p*(\alpha,\beta,\mu)$. In this paper we obtain for these classes sharp results concerning coefficient estimates, disortion theorems, closure theorems, Hadamard products and some distortion theorems for the fractional calculus.
ON CERTAIN SUBCLASS OF STARLIKE FUNCTIONS OF ORDER ${\alpha}\cdot$ AND TYPE $\beta$
Aouf, M.K. The Youngnam Mathematical Society Korea 1989 East Asian mathematical journal Vol.5 No.1
Let $S_o*({\alpha},{\beta},{\mu})$ denote the class of functions $f(z)=a_1z-{\sum}{\limit}^{\infty}_{n=2}\;a_nz^n$ analytic in the unit disc $U=\{z:{\mid}z{\mid}<1\}$ and satisfying the condition $${\mid}\frac{\frac{zf'(z)}{f(z)}-1}{(1+\mu)\;\beta(\frac{zf'(z)}{f(z)}-\alpha)-(\frac{zf'(z)}{f(z)}-1)}\mid<1$$ for some $\alpha(0{\leq}{\alpha}<1),\;{\beta}(0<{\beta}{\leq}1),\;{\mu}(0{\leq}{\mu}{\leq}1)$ and for all $z{\in}U$. And it is the purpose of this paper to show a necessary and sufficient condition for the class $S_o*({\alpha},{\beta},{\mu})$, some results for the Hadamard products of two functions f(z) and g(z) in the class $S_o*({\alpha},{\beta},{\mu})$, the distortion theorem and the distortion theorems for the fractional calculus.
Some subclasses of multivalent functions involving a certain linear operator
M. K. Aouf,H. Silverman,H. M. Srivastava 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.14 No.2
In this paper, we investigate the various important properties and characteristics of the subclasses Sa,c(λ;p,A,B) and Ta,c(λ;p,A,B) of p-valent functions defined by means of certain linear operators. We first establish an inclusion relation for the class Sa,c(λ;p,A,B). We then derive many results for the modied Hadamard products of functions belonging to the class Ta,c(λ;p,A,B). Finally, several applications involving an integral operator and certain fractional calculus operators are also considered.
AN APPLICATION OF CERTAIN LINEAR OPERATOR
Aouf, M.K.,Hossen, H.M.,Lashin, A.Y. Korean Mathematical Society 2000 대한수학회보 Vol.37 No.4
The object of the present paper is to give an application of a linear operator $L_p(a, c)$ defined by means of a Hadamard product (or convolution) to a Miller and Mocanu’s theorem.