Displaying similar documents to “Coefficient inequality for transforms of parabolic starlike and uniformly convex functions”

Coefficient inequality for a function whose derivative has a positive real part of order α

Deekonda Vamshee Krishna, Thoutreddy Ramreddy (2015)

Mathematica Bohemica

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The objective of this paper is to obtain sharp upper bound for the function f for the second Hankel determinant | a 2 a 4 - a 3 2 | , when it belongs to the class of functions whose derivative has a positive real part of order α ( 0 α < 1 ) , denoted by R T ( α ) . Further, an upper bound for the inverse function of f for the nonlinear functional (also called the second Hankel functional), denoted by | t 2 t 4 - t 3 2 | , was determined when it belongs to the same class of functions, using Toeplitz determinants.

On certain subclasses of analytic functions associated with the Carlson–Shaffer operator

Jagannath Patel, Ashok Kumar Sahoo (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class R λ ( a , c , A , B ) of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass R ˜ λ ( a , c , A , B ) of R λ ( a , c , A , B ) and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

Existence results for a class of nonlinear parabolic equations with two lower order terms

Ahmed Aberqi, Jaouad Bennouna, M. Hammoumi, Mounir Mekkour, Ahmed Youssfi (2014)

Applicationes Mathematicae

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We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧ ( e β u - 1 ) / t - d i v ( | u | p - 2 u ) + d i v ( c ( x , t ) | u | s - 1 u ) + b ( x , t ) | u | r = f in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ ( e β u - 1 ) ( x , 0 ) = ( e β u - 1 ) ( x ) in Ω. with s = (N+2)/(N+p) (p-1), c ( x , t ) ( L τ ( Q T ) ) N , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), b ( x , t ) L N + 2 , 1 ( Q T ) and f ∈ L¹(Q).

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): u t - Δ u = | u | p - 1 u + b | u | q . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C ( T - t ) - 1 / ( p - 1 ) | | u ( t ) | | C ( T - t ) - 1 / ( p - 1 ) . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality u t - u x x u p . More general inequalities of the form u t - u x x f ( u ) with, for instance, f ( u ) = ( 1 + u ) l o g p ( 1 + u ) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

Region of variability for functions with positive real part

Saminathan Ponnusamy, Allu Vasudevarao (2010)

Annales Polonici Mathematici

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For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let γ , β denote the class of all analytic functions P in the unit disk with P(0) = 1 and R e ( e i γ P ( z ) ) > β c o s γ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability V ( z , λ ) for 0 z P ( ζ ) d ζ when P ranges over the class ( λ ) = P γ , β : P ' ( 0 ) = 2 ( 1 - β ) λ e - i γ c o s γ . As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

Generalized problem of starlikeness for products of close-to-star functions

Jacek Dziok (2013)

Annales Polonici Mathematici

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We consider functions of the type F ( z ) = z j = 1 n [ f j ( z ) / z ] a j , where a j are real numbers and f j are β j -strongly close-to-starlike functions of order α j . We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type u t - ( - Δ ) β / 2 u - V ( x ) u + h ( x , t ) u p , where ( - Δ ) β / 2 , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying V ( x ) a | x | - b , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space

Xing-Tang Dong, Ze-Hua Zhou (2013)

Studia Mathematica

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We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators T f T g on the harmonic Bergman space is equal to a Toeplitz operator T h , then the product T g T f is also the Toeplitz operator T h , and hence T f commutes with T g . From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator.

Some properties for α -starlike functions with respect to k -symmetric points of complex order

H. E. Darwish, A. Y. Lashin, S. M. Sowileh (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In the present work, we introduce the subclass 𝒯 γ , α k ( ϕ ) , of starlike functions with respect to k -symmetric points of complex order γ ( γ 0 ) in the open unit disc . Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

The regularity of the positive part of functions in L 2 ( I ; H 1 ( Ω ) ) H 1 ( I ; H 1 ( Ω ) * ) with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let u L 2 ( I ; H 1 ( Ω ) ) with t u L 2 ( I ; H 1 ( Ω ) * ) be given. Then we show by means of a counter-example that the positive part u + of u has less regularity, in particular it holds t u + L 1 ( I ; H 1 ( Ω ) * ) in general. Nevertheless, u + satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang Phung, Gengsheng Wang (2013)

Journal of the European Mathematical Society

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This paper presents a new observability estimate for parabolic equations in Ω × ( 0 , T ) , where Ω is a convex domain. The observation region is restricted over a product set of an open nonempty subset of Ω and a subset of positive measure in ( 0 , T ) . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

The essential spectrum of Toeplitz tuples with symbols in H + C

Jörg Eschmeier (2013)

Studia Mathematica

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Let H²(D) be the Hardy space on a bounded strictly pseudoconvex domain D ⊂ ℂⁿ with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with H -symbol, we show that a Toeplitz tuple T f = ( T f , . . . , T f ) L ( H ² ( σ ) ) m with symbols f i H + C is Fredholm if and only if the Poisson-Szegö extension of f is bounded away from zero near the boundary of D. Corresponding results are obtained for the case of Bergman spaces. Thus we extend results of McDonald (1977) and...

Convolution conditions for bounded α -starlike functions of complex order

A. Y. Lashin (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let A be the class of analytic functions in the unit disc U of the complex plane with the normalization f ( 0 ) = f ' ( 0 ) - 1 = 0 . We introduce a subclass S M * ( α , b ) of A , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class S M * ( n , α , b ) ( n 0 ) related to S M * ( α , b ) is also considered under the same conditions. Among other things, we find convolution conditions for a function f A to belong to the class S M * ( α , b ) . Several properties of the class S M * ( n , α , b ) are investigated. ...

Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

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We study Toeplitz operators T a with radial symbols in weighted Bergman spaces A μ p , 1 < p < ∞, on the disc. Using a decomposition of A μ p into finite-dimensional subspaces the operator T a can be considered as a coefficient multiplier. This leads to new results on boundedness of T a and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of T a for a satisfying an assumption on the positivity of certain...

Properties of functions concerned with Caratheodory functions

Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let 𝒫 n denote the class of analytic functions p ( z ) of the form p ( z ) = 1 + c n z n + c n + 1 z n + 1 + in the open unit disc 𝕌 . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for p ( z ) concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

The generalized Toeplitz operators on the Fock space F α 2

Chunxu Xu, Tao Yu (2024)

Czechoslovak Mathematical Journal

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Let μ be a positive Borel measure on the complex plane n and let j = ( j 1 , , j n ) with j i . We study the generalized Toeplitz operators T μ ( j ) on the Fock space F α 2 . We prove that T μ ( j ) is bounded (or compact) on F α 2 if and only if μ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for T μ ( j ) to be in the Schatten p -class for 1 p < .

On Pólya's Theorem in several complex variables

Ozan Günyüz, Vyacheslav Zakharyuta (2015)

Banach Center Publications

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Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let f ( z ) = k = 0 a k z - k - 1 be its Taylor expansion at ∞, and H s ( f ) = d e t ( a k + l ) k , l = 0 s the sequence of Hankel determinants. The classical Pólya inequality says that l i m s u p s | H s ( f ) | 1 / s ² d ( K ) , where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations D u K , where u : Ω n m is a Lipschitz map and K is a bounded set in m × n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of D u and second we replace Gromov’s P −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from 0 and show that if f 𝒫 H 0 ( α ) and F 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections...