In this manuscript, we deal with three classes of quotient functions having fixed second coeffici... more In this manuscript, we deal with three classes of quotient functions having fixed second coefficient described on open unit disk. The radius of strongly starlikeness, lemniscate starlikeness, lune starlikeness, parabolic starlikeness, sine starlikeness, exponential starlikeness and several other radius estimates for such classes are examined. Relevant relations of obtained radius estimates with the existing estimates are also discussed.
The leukotrienes are potent inflammatory mediators, which may have a role in inflammatory disease... more The leukotrienes are potent inflammatory mediators, which may have a role in inflammatory diseases such as allergic rhinitis, inflammatory bowl disease and asthma. Zafirlukast, a cysteinyl leukotriene receptor antagonist, is claimed to be effective in asthma. However, it is not known whether ...
Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.
In this paper, we introduce a new generalized class of analytic functions involving the Mittag-Le... more In this paper, we introduce a new generalized class of analytic functions involving the Mittag-Leffler operator and Bazilevic̆ functions. We examine inclusion properties, radius problems and an application of the generalized Bernardi-Libera-Livingston integral operator for this function class.
Let $\phi$ be a normalized convex function defined on open unit disk $\mathbb{D}$. For a unified ... more Let $\phi$ be a normalized convex function defined on open unit disk $\mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f'(z)+ \alpha z f''(z) \prec \phi(z)$ for all $z\in \mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.
The Bohr radius for a class $${\mathcal {G}}$$ G consisting of analytic functions $$f(z)=\sum _{n... more The Bohr radius for a class $${\mathcal {G}}$$ G consisting of analytic functions $$f(z)=\sum _{n=0}^{\infty }a_nz^n$$ f ( z ) = ∑ n = 0 ∞ a n z n in unit disc $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ D = { z ∈ C : | z | < 1 } is the largest $$r^*$$ r ∗ such that every function f in the class $${\mathcal {G}}$$ G satisfies the inequality $$\begin{aligned} d\left( \sum _{n=0}^{\infty }|a_nz^n|, |f(0)|\right) = \sum _{n=1}^{\infty }|a_nz^n|\le d(f(0), \partial f({\mathbb {D}})) \end{aligned}$$ d ∑ n = 0 ∞ | a n z n | , | f ( 0 ) | = ∑ n = 1 ∞ | a n z n | ≤ d ( f ( 0 ) , ∂ f ( D ) ) for all $$|z|=r \le r^*$$ | z | = r ≤ r ∗ , where d is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions f satisfying differential subordination relations $$zf'(z)/f(z) \prec h(z)$$ z f ′ ( z ) / f ( z ) ≺ h ( z ) and $$f(z)+\beta z f'(z)+\gamma z^2 f''(z)\prec h(z)$$ f ( z ) + β z f ′ ( z ) + γ z 2 f ′ ′ ( z ) ≺ ...
In this article, we wish to establish some first order differential subordination relations for c... more In this article, we wish to establish some first order differential subordination relations for certain Carathéodory functions with nice geometrical properties. Moreover, several implications are determined so that the normalized analytic function belongs to various subclasses of starlike functions.
Some sufficient conditions on certain constants which are involved in some first, second and thir... more Some sufficient conditions on certain constants which are involved in some first, second and third order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential function and Janowski function are obtained so that the analytic function p normalized by the condition p(0) = 1, is subordinate to Janowski function. The admissibility conditions for Janowski function are used as a tool in the proof of the results. As application, several sufficient conditions are also computed for Janowski starlikeness.
The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr ra... more The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order α . The Bohr radius is also determined for a generalized class of q-Janowski starlike and q-Janowski convex functions with negative coefficients.
For fi belonging to various subclasses of univalent functions, we investigate the product given b... more For fi belonging to various subclasses of univalent functions, we investigate the product given by h(z) = z n i=1 (f i (z)/z) γ i. The largest radius ρ is determined such that h(ρz)/ρ is starlike of order β, 0 ≤ β < 1 or to belong to other subclasses of univalent functions. We also determine the sharp radius of starlikeness of order β and other radius for the convolution f * g of two starlike functions f, g.
Several radii problems are considered for functions f (z) = z + a2z2 + ... with fixed second coef... more Several radii problems are considered for functions f (z) = z + a2z2 + ... with fixed second coefficient a2. For 0 ? ? < 1, sharp radius of starlikeness of order ? for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order ? for uniformly convex functions, and sharp radius of strong-starlikeness of order ? for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.
Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.
Page 1. Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Qu... more Page 1. Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HQM2010) Editors: D. Minda, S. Ponnusamy, and N. Shanmugalingam J. Analysis Volume 18 (2010), 18 Convolutions of Certain Analytic Functions ...
For a normalized analytic functionfdefined on the unit disc𝔻, letϕ(f,f′,f′′;z)be a function of po... more For a normalized analytic functionfdefined on the unit disc𝔻, letϕ(f,f′,f′′;z)be a function of positive real part in𝔻,ψ(f,f′,f′′;z)need not have that property in𝔻, andχ=ϕ+ψ. For certain choices ofϕandψ, a sharp radius constantρis determined,0<ρ<1, so thatχ(ρz)/ρmaps𝔻onto a specified region in the right half-plane.
Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.
A normalized analytic function f defined on the open unit disk in the complex plane is in the cla... more A normalized analytic function f defined on the open unit disk in the complex plane is in the class SL if zf'(z)/f(z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w^2 - 1| < 1. In the present investigation, the SL-radii for certain well-known classes of functions are obtained. Radius problems associated with the left-half plane are also investigated for these classes
In this manuscript, we deal with three classes of quotient functions having fixed second coeffici... more In this manuscript, we deal with three classes of quotient functions having fixed second coefficient described on open unit disk. The radius of strongly starlikeness, lemniscate starlikeness, lune starlikeness, parabolic starlikeness, sine starlikeness, exponential starlikeness and several other radius estimates for such classes are examined. Relevant relations of obtained radius estimates with the existing estimates are also discussed.
The leukotrienes are potent inflammatory mediators, which may have a role in inflammatory disease... more The leukotrienes are potent inflammatory mediators, which may have a role in inflammatory diseases such as allergic rhinitis, inflammatory bowl disease and asthma. Zafirlukast, a cysteinyl leukotriene receptor antagonist, is claimed to be effective in asthma. However, it is not known whether ...
Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.
In this paper, we introduce a new generalized class of analytic functions involving the Mittag-Le... more In this paper, we introduce a new generalized class of analytic functions involving the Mittag-Leffler operator and Bazilevic̆ functions. We examine inclusion properties, radius problems and an application of the generalized Bernardi-Libera-Livingston integral operator for this function class.
Let $\phi$ be a normalized convex function defined on open unit disk $\mathbb{D}$. For a unified ... more Let $\phi$ be a normalized convex function defined on open unit disk $\mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f'(z)+ \alpha z f''(z) \prec \phi(z)$ for all $z\in \mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.
The Bohr radius for a class $${\mathcal {G}}$$ G consisting of analytic functions $$f(z)=\sum _{n... more The Bohr radius for a class $${\mathcal {G}}$$ G consisting of analytic functions $$f(z)=\sum _{n=0}^{\infty }a_nz^n$$ f ( z ) = ∑ n = 0 ∞ a n z n in unit disc $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ D = { z ∈ C : | z | < 1 } is the largest $$r^*$$ r ∗ such that every function f in the class $${\mathcal {G}}$$ G satisfies the inequality $$\begin{aligned} d\left( \sum _{n=0}^{\infty }|a_nz^n|, |f(0)|\right) = \sum _{n=1}^{\infty }|a_nz^n|\le d(f(0), \partial f({\mathbb {D}})) \end{aligned}$$ d ∑ n = 0 ∞ | a n z n | , | f ( 0 ) | = ∑ n = 1 ∞ | a n z n | ≤ d ( f ( 0 ) , ∂ f ( D ) ) for all $$|z|=r \le r^*$$ | z | = r ≤ r ∗ , where d is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions f satisfying differential subordination relations $$zf'(z)/f(z) \prec h(z)$$ z f ′ ( z ) / f ( z ) ≺ h ( z ) and $$f(z)+\beta z f'(z)+\gamma z^2 f''(z)\prec h(z)$$ f ( z ) + β z f ′ ( z ) + γ z 2 f ′ ′ ( z ) ≺ ...
In this article, we wish to establish some first order differential subordination relations for c... more In this article, we wish to establish some first order differential subordination relations for certain Carathéodory functions with nice geometrical properties. Moreover, several implications are determined so that the normalized analytic function belongs to various subclasses of starlike functions.
Some sufficient conditions on certain constants which are involved in some first, second and thir... more Some sufficient conditions on certain constants which are involved in some first, second and third order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential function and Janowski function are obtained so that the analytic function p normalized by the condition p(0) = 1, is subordinate to Janowski function. The admissibility conditions for Janowski function are used as a tool in the proof of the results. As application, several sufficient conditions are also computed for Janowski starlikeness.
The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr ra... more The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order α . The Bohr radius is also determined for a generalized class of q-Janowski starlike and q-Janowski convex functions with negative coefficients.
For fi belonging to various subclasses of univalent functions, we investigate the product given b... more For fi belonging to various subclasses of univalent functions, we investigate the product given by h(z) = z n i=1 (f i (z)/z) γ i. The largest radius ρ is determined such that h(ρz)/ρ is starlike of order β, 0 ≤ β < 1 or to belong to other subclasses of univalent functions. We also determine the sharp radius of starlikeness of order β and other radius for the convolution f * g of two starlike functions f, g.
Several radii problems are considered for functions f (z) = z + a2z2 + ... with fixed second coef... more Several radii problems are considered for functions f (z) = z + a2z2 + ... with fixed second coefficient a2. For 0 ? ? < 1, sharp radius of starlikeness of order ? for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order ? for uniformly convex functions, and sharp radius of strong-starlikeness of order ? for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.
Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.
Page 1. Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Qu... more Page 1. Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HQM2010) Editors: D. Minda, S. Ponnusamy, and N. Shanmugalingam J. Analysis Volume 18 (2010), 18 Convolutions of Certain Analytic Functions ...
For a normalized analytic functionfdefined on the unit disc𝔻, letϕ(f,f′,f′′;z)be a function of po... more For a normalized analytic functionfdefined on the unit disc𝔻, letϕ(f,f′,f′′;z)be a function of positive real part in𝔻,ψ(f,f′,f′′;z)need not have that property in𝔻, andχ=ϕ+ψ. For certain choices ofϕandψ, a sharp radius constantρis determined,0<ρ<1, so thatχ(ρz)/ρmaps𝔻onto a specified region in the right half-plane.
Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.
A normalized analytic function f defined on the open unit disk in the complex plane is in the cla... more A normalized analytic function f defined on the open unit disk in the complex plane is in the class SL if zf'(z)/f(z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w^2 - 1| < 1. In the present investigation, the SL-radii for certain well-known classes of functions are obtained. Radius problems associated with the left-half plane are also investigated for these classes
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