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.
Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition. We exploit the first order differential subordination theory to get several sufficient conditions for function satisfying several differential subordinations to be a Janowski function with positive real part. As applications, we obtain sufficient conditions for normalized analytic functions to be Janowski starlike functions.
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.
In this paper, we obtain some application of first-order differential subordination, superordination and sandwich-type results involving operator for certain normalized $p$-valent analytic functions. Further, properties of $p$-valent functions such as; $λ$-spirallike and $λ$-Robertson of complex order are considered.
Differential subordination and superordination preserving properties for univalent functions in the open unit disk with an operator involving generalized Bessel functions are derived. Some particular cases involving trigonometric functions of our main results are also pointed out.
We introduce and study the following special type of differential subordination implication: \begin{equation}\label{abs} p(z)Q(z)+\frac{zp'(z)}{βp(z)+α}\prec h(z)\quad\Rightarrow p(z)\prec h(z), \end{equation} which generalizes the Briot-Bouquet differential subordination, where $Q(z)$ is analytic and $0\neqβ,α\in\mathbb{C}.$ Further, some special cases of our result are also discussed. Finally, analogues of open door lemma and integral existence theorem with applications to univalent functions are obtained.
In the present paper, we derive the third-order differential subordination and superordination results for some analytic univalent functions defined in the unit disc. These results are associated with generalized Struve functions and are obtained by considering suitable classes of admissible functions. As a consequence, the dual problems which yield the sandwich type relations are presented.
The theory of first-order differential subordination developed by Miller and Mocanu was recently extended to functions with fixed initial coefficient by R. M. Ali, S. Nagpal and V. Ravichandran [Second-order differential subordination for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc. (2) 34 (2011), 611--629] and applied to obtain several generalization of classical results in geometric function theory. In this paper, further applications of this subordination theory is given. In particular, several sufficient conditions related to starlikeness, convexity, clo…
By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions is evident in the results obtained.
For $0\leqα\leq 1,$ let $H_α(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form \begin{equation*} H_α(p(z),p(z)Θ(z)+zp'(z)Φ(z))\prec h(z)\Rightarrow p(z)\prec h(z), \end{equation*} where $Φ,\;Θ$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.
Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several interesting examples are given. The results obtained are applied to the classes of convex and starlike functions with fixed second coefficient.
Let $q_1$ and $q_2$ belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions $p$ to satisfy the double subordination chain $q_1(z)\prec p(z)\prec q_2(z)$. The differential sandwich-type result obtained is applied to normalized univalent functions and to $Φ$-like functions.
We consider the class $\mathcal{S}^*(q_c)$ of normalized starlike functions $f$ analytic in the open unit disk $|z|<1$ that satisfying the inequality \begin{equation*} \left|\left(\frac{zf'(z)}{f(z)}\right)^2-1\right|<c \quad (0<c\leq1). \end{equation*} In this article, we present some subordination relations and these relations are then used to obtain some corollaries for some subclass of analytic functions.
In this article we study the inclusion properties of the Bessel-Struve kernel functions in the Janowski class. In particular, we find the conditions for which the Bessel-Struve kernel functions maps the unit disk to right half plane. An open problem in this aspect are also given. The third order differential subordination involving the Bessel-Struve kernel is also considered. The results are derived by defining suitable classes of admissible functions. One of the recurrence relation of the Bessel-Struve kernel play an important role to derive the main results.
In this paper our aim is to present some subordination and superordination results, by using an operator, which involves the normalized form of the generalized Bessel functions of first kind. These results are obtained by investigating some appropriate classes of admissible functions. We obtain also some sandwich-type results and we point out various known or new special cases of our main results.
In this article, by making use of the linear operator introduced and studied by Srivastava and Attiya \cite{srivastava1}, suitable classes of admissible functions are investigated and the dual properties of the third-order differential subordinations are presented. As a consequence, various sandwich-type theorems are established for a class of univalent analytic functions involving the celebrated Srivastava-Attiya transform. Relevant connections of the new results are pointed out.
Sharp estimates on $β$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever $1+βz p'(z), \,\,1+βz p'(z)/p(z), \,\,\mbox{or}\,\,1+βz p'(z)/p^{2}(z)$ is subordinate to $\sqrt{1+z}$. Our results provide sharp version of previously known results.
Let $h$ be a non vanishing convex univalent function and $p$ be an analytic function in $\mathbb{D}$. We consider the differential subordination $$ψ_i(p(z), z p'(z)) \prec h(z)$$ with the admissible functions in consideration as $ψ_1:=(βp(z)+γ)^{-α}\left(\tfrac{(βp(z)+γ)}{β(1-α)}+ z p'(z)\right)$ and $ψ_2:=\tfrac{1}{\sqrt{γβ}}\arctan\left(\sqrt{\tfracβγ}p^{1-α}(z)\right)+\left(\tfrac{1-α}{βp^{2 (1-α)}(z)+γ}\right)\tfrac{z p'(z)}{p^α(z)}$. The objective of this paper is to find the dominants, preferably the best dominant(say $q$) of the solution of the above differential subordination satisfyin…
For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$ satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination of first or second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to prove the main results.
Let $\mathcal{A}$ be the set of all analytic functions $f$ defined in the open unit disk $\mathbb{D}$ and satisfying $f(0)=f'(0)-1=0$. In this paper, we consider the function $φ_{\scriptscriptstyle {Ne}}(z):=1+z-z^3/3$, which maps the unit circle $\{z:|z|=1\}$ onto a $2$-cusped curve called nephroid given by $\left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0$, and the function class $\mathcal{S}^*_{Ne}$ defined as \begin{align*} \mathcal{S}^*_{Ne}:=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\precφ_{\scriptscriptstyle {Ne}}(z)\right\}, \end{align*} where $\prec$ denotes subordination. We …
For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian derivative Q_{SD}:=z^2((f"(z)/f'(z))'-(f"(z)/f'(z))^2/2)$. These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.
In the present paper, we define a new general subclass of bi-univalent functions involving a differential operator in the open unit disk U. For this purpose, we use the Faber polynomial expansions. Several connections to some of the earlier known results are also pointed out.
The conditions on $A$, $B$, $β$ and $γ$ are obtained for an analytic function $p$ defined on the open unit disc $\mathbb{D}$ and normalized by $p(0)=1$ to be subordinate to $(1+Az)/(1+Bz)$, $-1\leq B<A \leq 1$ when $p(z)+ zp'(z)/(βp(z)+γ)$ is subordinate to $e^{z}$. The conditions on these parameters are derived for the function $p$ to be subordinate to $\sqrt{1+z}$ or $e^{z}$ when $p(z)+ zp'(z)/(βp(z)+γ)$ is subordinate to $(1+Az)/(1+Bz)$. The conditions on $β$ and $γ$ are determined for the function $p$ to be subordinate to $e^{z}$ when $p(z)+ zp'(z)/(βp(z)+γ)$ is subordinate to $\sqrt{1+z}$…
Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*} \mathcal{R}(α,β):=\left\{ f\in \mathcal{A}: {\rm Re}\left\{f'(z)+\frac{1+e^{iα}}{2}zf''(z)\right\}>β,\, |z|<1\right\} \end{equation*} and \begin{equation*} \mathcal{L}_α(b):=\left\{f\in\mathcal{A}:\left|f'(z) +\frac{1+e^{iα}}{2}zf''(z)-b\right|< b,\, |z|<1 \right\}, \end{equation*} where $-π<α\leq π$, $0\leq β<1$ and $b>1/2$. We show that if $f\in \mathcal{R}(α,β)$, then ${\rm Re}\{f'(z)\}$ and ${\rm Re}\{f(z)/z\}$ are greater t…
Some differential implications of classical Marx-Strohhäcker theorem are extended for multivalent functions. These results are also generalized for functions with fixed second coefficient by using the theory of first order differential subordination which in turn, corrects the results of Selvaraj and Stelin [On multivalent functions associated with fixed second coefficient and the principle of subordination, Int. J. Math. Anal. {\bf 9} (2015), no.~18, 883--895].
Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz \emph{et al.} [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. {\bf 38} (2015), no.~3, 1255--1280]. These results are obtaine…
Let $\mathcal{A}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+\sum_{k=n+1}^\infty a_kz^k,n\in\mathbb{N}$ and let \begin{align*} Ω_n:=\left\{f\in\mathcal{A}_n:\left|zf'(z)-f(z)\right|<\frac{1}{2},\; z\in\mathbb{D}\right\}. \end{align*} We make use of differential subordination technique to obtain sufficient conditions for the class $Ω_n$, and then employ these conditions to construct functions which involve double integrals and members of $Ω_n$. We also consider a subclass $\widehatΩ_n\subsetΩ_n$ and obtain subordination results for members of $\widehatΩ_n$ besides a necessa…
Assume that $Δ$ is the open unit disk in the complex plane and $\mathcal{A}$ is the class of normalized analytic functions in $Δ$. In this paper we introduce and study the class \begin{equation*} \mathcal{BS}(α):=\left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1-αz^2}, \, z\inΔ\right\}, \end{equation*} where $0\leqα\leq1$ and $\prec$ is the subordination relation. Some properties of this class like differential subordination, coefficients estimates and Fekete-Szegö inequality associated with the $k$-th root transform are considered.
In this paper we introduce and study two new subclasses Σ_{λμ mp}(α,β)$ and $Σ^{+}_{λμmp}(α,β)$ of meromorphically multivalent functions which are defined by means of a new differential operator. Some results connected to subordination properties, coefficient estimates, convolution properties, integral representation, distortion theorems are obtained. We also extend the familiar concept of $% (n,δ)-$neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions.
In this short note we employ the Briot-Bouquet differential subordination to determine the best possible inclusion relation within a certain family of analytic functions defined by the Salagean derivative.
Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.
Let $ϕ$ 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 $f'(z)+ αz f''(z) \prec ϕ(z)$ 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 object of the present paper is to study certain properties and characteristics of the operator $Q_{p,β}^α$defined on p-valent analytic function by using technique of differential subordination.We also obtained result involving majorization problems by applying the operator to p-valent analytic function.Relevant connection of the the result are presented here with those obtained by earlier worker are pointed out.
We introduce a class of analytic functions subordinate to the function $1+\sinh \left( z\right) $ and obtain various necessary and sufficient conditions for functions to be in the class. These conditions mainly comprise of the coefficient inequalities involving convolution. Further, we have obtained sharp five initial coefficients, a conjecture for the general nth coefficient and the third Hankel determinant bounds for the functions in this class. Also derived certain differential subordination implication results involving $1+\sinh \left( z\right)$.
In this paper, we show that the Carathéodory function $φ_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3$ maps the open unit disk $\mathbb{D}$ onto the interior of the nephroid, a $2$-cusped kidney-shaped curve, \begin{align*} \left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0, \end{align*} and introduce new Ma-Minda type function classes $\mathcal{S}^*_{Ne}$ and $\mathcal{C}_{Ne}$ associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient b…
Since the nonlinear integral transforms $J_α[f](z) = \int_{0}^{z}(f'(u))^α du$ and $I_α[f](z) =\int_0^z (f(u)/u)^α du$ with a complex number $α$ have been introduced, a great number of studies were dedicated to deriving sufficient conditions for univalence on the unit disk. On the other hand, little is known about the conditions that $J_α[f]$ or $I_α[f]$ produces a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper we discuss quasiconformal extendibility of the integral transforms $J_α[f]$ and $I_α[f]$ for holomorphic funct…
In this paper our aim is to determine the radii of univalence, starlikeness and convexity of the normalized regular Coulomb wave functions for two different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for regular Coulomb wave functions, and properties of zeros of the regular Coulomb wave functions and their derivatives. Moreover, by using the technique of differential subordinations we present some conditions on the parameters of the regular Coulomb wave function in order to have a starlike normalized form. In addition, by using the E…
The Bohr radius for a class $\mathcal{G}$ consisting of analytic functions $f(z)=\sum_{n=0}^{\infty}a_nz^n$ in unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ is the largest $r^*$ such that every function $f$ in the class $\mathcal{G}$ satisfies the inequality \begin{equation*} d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right) = \sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})) \end{equation*} for all $|z|=r \leq r^*$, 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 subor…
In this paper consisting of two parts, we study the integral of a logarithmic differential form on a compact semi-algebraic set in R^n or C^n. In Part I, we prove the convergence of the integral when the semi-algebraic set satisfies allowability (or admissibility), a condition on the dimension of the intersection of the set and the pole divisor of the differential form.
