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We obtain various estimates of the life-time of two-dimensional minimal tubes in R^3 by potential theory methods.

We prove three sharp estimates for the generalized Zalcman coefficient functional: one for the Hurwitz class, another for the Noshiro-Warschawski class, and yet another for the functions in the closed convex hull of convex univalent functions. In each case the extremal functions are identified. We also observe that an asymptotic version of the Zalcman conjecture is true.

We prove three sharp estimates for the generalized Zalcman coefficient functional: one for the Hurwitz class, another for the Noshiro-Warschawski class, and yet another for the functions in the closed convex hull of convex univalent functions. In each case the extremal functions are identified. We also observe that an asymptotic version of the Zalcman conjecture is true.

In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class.

Estimates for initial coefficients of Taylor-Maclaurin series of bi-univalent functions belonging to certain classes defined by subordination are obtained. Our estimates improve upon the earlier known estimates for second and third coefficient. The bound for the fourth coefficient is new. In addition, bound for the fifth coefficient is obtained for bi-starlike and strongly bi-starlike functions of order $\rho$ and $\beta$ respectively.

We give some coefficient bounds and distortion theorems for a subclass of univalent functions in the unit disk, and defined using the S\^{a}l\^{a}gean differential operator. The results generalize and unify some well known results for several subclasses of univalent functions defined on the unit disk having form $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$, normalized such that $f(0)=0$ and $f^\prime(0)=1$.

In this paper we extend the concept of bi-univalent to the class of meromorphic functions. We propose to investigate the coefficient estimates for two classes of meromorphic bi-univalent functions. Also, we find estimates on the coefficients |b0| and |b1| for functions in these new classes. Some interesting remarks and applications of the results presented here are also discussed.

In this sequel to the recent work (see Azizi et al., 2015), we investigate a subclass of analytic and bi-univalent functions in the open unit disk. We obtain bounds for initial coefficients, the Fekete-Szeg\"o inequality and the second Hankel determinant inequality for functions belonging to this subclass. We also discuss some new and known special cases, which can be deduced from our results.

Let ${\mathcal U}(\lambda)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |

Given a set $\mathfrak S$ of conformal maps of the unit disk $\mathbb D$ into itself that is closed under composition, we address the question whether $\mathfrak S$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$, $w_0=\mathsf{id}_{\mathbb D}$, where the control functions $t\mapsto G_t$ form a convex cone $\mathcal M_{\mathfrak S}$. For the set of all conformal $\varphi:\mathbb D\to\mathbb D$ with $\varphi(0)=0$, $\varphi'(0)>0$, an affirmative answer to this question is the essence of Loewner's well-known Parametric Representation of univalent functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of conformal self-maps defined by their boundary regular fixed points and, in part of the cases, establish their Loewner-type representability.

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (\lambda)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1|

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (\lambda)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1|

Book Source: Digital Library of India Item 2015.189533 dc.contributor.author: Surendra Prasad Dwivedi dc.date.accessioned: 2015-07-07T23:12:40Z dc.date.available: 2015-07-07T23:12:40Z dc.date.digitalpublicationdate: 2005-08-27 dc.identifier.barcode: 1990010086537 dc.identifier.origpath: /rawdataupload/upload/0086/537 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/189533 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 124 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.title: Certain Classes Of Univalent Funtions And Generalizations Of Functions With Bounded Boundary Rotations

Let $Co(\alpha)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(\alpha)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(\alpha)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in Co(\alpha)$ whenever $f''(0)$ is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in $Co(\alpha)$ belong to the $H^p$ space for $p

Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of the Virasoro algebra in the space of univalent functions.

It turns out that complex geodesics in Teichm\"uller spaces with respect to their invariant metrics are intrinsically connected with variational calculus for univalent functions. We describe this connection and show how geometric features associated to these metrics and geodesics provide deep distortion results for various classes of functions with quasiconformal extensions and create new phenomena which do not appear in the classical geometric function theory.

The article deals with the class ${\mathcal F}_{\alpha }$ consisting of non-vanishing functions $f$ that are analytic and univalent in $\ID$ such that the complement $\IC\backslash f(\ID) $ is a convex set, $f(1)=\infty ,$ $f(0)=1$ and the angle at $\infty $ is less than or equal to $\alpha \pi ,$ for some $\alpha \in (1,2]$. Related to this class is the class $CO(\alpha)$ of concave univalent mappings in $\ID$, but this differs from ${\mathcal F}_{\alpha }$ with the standard normalization $f(0)=0=f'(0)=1.$ A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for $CO(2)$ settled by Avkhadiev et al. \cite{Avk-Wir-04}. Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for $CO(\alpha)$ is also presented.

The article deals with the class ${\mathcal F}_{\alpha }$ consisting of non-vanishing functions $f$ that are analytic and univalent in $\ID$ such that the complement $\IC\backslash f(\ID) $ is a convex set, $f(1)=\infty ,$ $f(0)=1$ and the angle at $\infty $ is less than or equal to $\alpha \pi ,$ for some $\alpha \in (1,2]$. Related to this class is the class $CO(\alpha)$ of concave univalent mappings in $\ID$, but this differs from ${\mathcal F}_{\alpha }$ with the standard normalization $f(0)=0=f'(0)=1.$ A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for $CO(2)$ settled by Avkhadiev et al. \cite{Avk-Wir-04}. Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for $CO(\alpha)$ is also presented.

Book Source: Digital Library of India Item 2015.196338 dc.contributor.author: Shyam Kishore Bajpai dc.date.accessioned: 2015-07-08T08:44:57Z dc.date.available: 2015-07-08T08:44:57Z dc.date.digitalpublicationdate: 0008-09-05 dc.identifier.barcode: 1990010087000 dc.identifier.origpath: /rawdataupload/upload/0087/000 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/196338 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 125 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: I I T Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Depart Of Mathmatic dc.title: Some Estimates Of Univalent And Multivalent Analytic Functions

Book Source: Digital Library of India Item 2015.196338 dc.contributor.author: Shyam Kishore Bajpai dc.date.accessioned: 2015-07-08T08:44:57Z dc.date.available: 2015-07-08T08:44:57Z dc.date.digitalpublicationdate: 0008-09-05 dc.identifier.barcode: 1990010087000 dc.identifier.origpath: /rawdataupload/upload/0087/000 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/196338 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 125 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: I I T Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Depart Of Mathmatic dc.title: Some Estimates Of Univalent And Multivalent Analytic Functions

We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients.

Let $\ID$ denote the open unit disk and $f:\,\ID\TO\BAR\IC$ be meromorphic and univalent in $\ID$ with the simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, let $f$ have the expansion $$f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|

Let $\ID$ denote the open unit disk and $f:\,\ID\TO\BAR\IC$ be meromorphic and univalent in $\ID$ with the simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, let $f$ have the expansion $$f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|

The two-functional conjecture says that if a function f analytic and univalent in the unit disk maximizes Re{L} and Re{M} for two continuous linear functionals L and M, L is not equal to cM for any c>0, then f is a rotation of the Koebe function. We use the Loewner differential equation to obtain sufficient conditions in the two-functional conjecture and compare the sufficient conditions with necessary conditions.

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, R. Singh and S. Singh (Coll. Math. 47(1982), 309-314) have considered some sufficient problems for f(z) to be univalent in U. The object of the present paper is to discuss some sufficient problems for f(z) to be some classes of analytic functions in U.

A conjecture of Bombieri states that the coefficients of a normalized univalent function $f$ should satisfy $$ \liminf_{f\to K} \frac{n-{\rm Re\,}a_n}{m-{\rm Re\,}a_m} = \min_{t\in{\mathbb R}} \, \frac{n\sin t -\sin(nt)}{m\sin t -\sin(mt)}, $$ when $f$ approaches the Koebe function $K(z)=\frac{z}{(1-z)^2}$. Recently, Leung disproved this conjecture for $n=2$ and for all $m\geq3$ and, also, for $n=3$ and for all odd $m\geq5$. Complementing his work we disprove it for all $m>n\geq2$ which are simultaneously odd or even and, also, for the case when $m$ is odd, $n$ is even and $n\leq \frac{m+1}{2}$. We mostly make use of trigonometry, but also employ Dieudonn\'e's criterion for the univalence of polynomials.

One of the classical problems concerns the class of analytic functions $f$ on the open unit disk $|z|

In the present work, we propose to investigate the second Hankel determinant inequalities for certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed.

We consider the Fekete-Szeg\"o problem with real parameter $\lambda$ for the class $Co(\alpha)$ of concave univalent functions.

In this work, we introduce two new subclasses S_{{\Sigma}_{m}}({\alpha},{\lambda}) and S_{{\Sigma}_{m}}(\b{eta},{\lambda}) of {\Sigma}_{m} consisting of analytic and m-fold symmetric bi-univalent functions in the open unit disk U. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for |a_{m+1}| and |a_{2m+1}|.

In this paper by making use of the generalized Bernardi - Libera - Livingston integral operator we introduce and study some new subclasses of univalent functions. Also we investigate the relations between those classes and the classes which are studied by Jin-Lin Liu.

In this paper, estimates for second and third MacLaurin coefficients of certain subclasses of bi-univalent functions in the open unit disk defined by convolution are determined, and certain special cases are also indicated. The main result extends and improve a recent one obtained by Srivastava et al.

We obtain the Kirillov vector fields on the set of functions $f$ univalent inside the unit disk, in terms of the Faber polynomials of $1/f(1/z)$. Our construction relies on the generating function for Faber polynomials.

Book Source: Digital Library of India Item 2015.192186 dc.contributor.author: Atul Kumar dc.date.accessioned: 2015-07-08T01:19:23Z dc.date.available: 2015-07-08T01:19:23Z dc.date.digitalpublicationdate: 0000-00-00 dc.identifier.barcode: 1990010090632 dc.identifier.origpath: /rawdataupload/upload/0090/632 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/192186 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 134 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Bishwaabhaarti Prakaashan Naagpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.title: A Unifiedan Input Output Stanalysis And Simulationof A Capaditorudy Of Industrial Water Pollution In India Study Of Certain Subclasses Of Univalent Analytic Functions

Book Source: Digital Library of India Item 2015.194314 dc.contributor.author: Vedanabhatla Srinivas dc.date.accessioned: 2015-07-08T04:17:39Z dc.date.available: 2015-07-08T04:17:39Z dc.date.digitalpublicationdate: 0000-00-00 dc.identifier.barcode: 1990010093125 dc.identifier.origpath: /rawdataupload/upload/0093/125 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/194314 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 267 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Iit Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Mathematics dc.title: A Study Of Certain Subclasses Of Univalent Analytic Functions

In this paper we introduce and investigate two new subclasses of the function class $\Sigma$ of bi-univalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-MacLaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several known or new consequences of these results are also pointed out.

In the present investigation, we consider two new subclasses N_{{\Sigma}}^{{\mu}}({\alpha},{\lambda}) and N_{{\Sigma}}^{{\mu}}({\beta},{\lambda}) of bi-univalent functions defined in the open unit disk U={z:|z|

In the present work, we propose to investigate the Fekete-Szeg\"o inequalities certain classes of analytic and bi-univalent functions defined by subordination. The results in the bounds of the third coefficient which improve many known results concerning different classes of bi-univalent functions. Some interesting applications of the results presented here are also discussed.

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP^1 we show that the open disk in CP^1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R=\frac{q}{p}. This solves a special case of the so-called Hermite-Biehler problem.

In this article we consider a family $\mathcal{C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\ID=\{z :|z|

Book Source: Digital Library of India Item 2015.192171 dc.contributor.author: Aparana Mishra dc.date.accessioned: 2015-07-08T01:18:59Z dc.date.available: 2015-07-08T01:18:59Z dc.date.digitalpublicationdate: 0000-00-00 dc.identifier.barcode: 1990010090619 dc.identifier.origpath: /rawdataupload/upload/0090/619 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/192171 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 231 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Bishwaabhaarti Prakaashan Naagpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.title: A Unifiedan Input Output Study Of Industrial Water Pollution In India Study Of Certain Subclasses Of Univalent Analytic Functions

For an analytic function $f(z)$ on the unit disk $|z|

The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk that was finally proved by de Branges some 70 years later, drifted many complex analysts attention to other subjects. Those who continued to explore de Branges method and push it as far as possible were not aware of where it may lead. Surprisingly enough, a paper by X. H. Dong that fell in our hands contained a way to tackle one of the problems of Bombieri on the behavior of the coefficients of univalent functions. We shall give an account of the history of the problem and a revised version of it.

We investigate the invariant metrics and complex geodesics in the universal Teichm\"{u}ller space and Teichm\"{u}ller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichm\"{u}ller metric. Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory.

In this article we consider functions $f$ meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions. This condition simplifies and generalizes known conditions. We present some typical problems of geometrical function theory and give elementary solutions in the case of the above functions.

Inspired by the recent works of Srivastava et al. (HMS-AKM-PG), Frasin and Aouf (BAF-MKA) and others (Ali-Ravi-Ma-Mina-class,Caglar-Orhan,Goyal-Goswami,Xu-HMS-AML,Xu-HMS-AMC), we propose to investigate the coefficient estimates for a general class of analytic and bi-univalent functions. Also, we obtain estimates on the coefficients |a2| and |a3| for functions in this new class. Some interesting remarks, corollaries and applications of the results presented here are also discussed.

We study the behavior of the initial coefficients of univalent functions under the Steiner symmetrization, and give some applications to functions of class \Sigma.

We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the L\"owner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system and calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case $\mathcal M_3$.

For a real constant $b,$ we give sharp estimates of $\log|f(z)/z|+b\arg[f(z)/z]$ for subclasses of normalized univalent functions $f$ on the unit disk.

A bi-univalent function is a univalent function defined on the unit disk with its inverse also univalent on the unit disk. Estimates for the initial coefficients are obtained for bi-univalent functions belonging to certain classes defined by subordination and relevant connections with earlier results are pointed out.