This paper aims to investigate several inequalities for parabolic trigonometric functions (PTF). We solve an open problem of proving Cusa-Huygens' inequality for PTF. Furthermore, we find indefinite integrals for PTF; consequently, we derive a tight approximation for a particular integral. Additionally, we develop Djokvie's type inequality for PTF and discover a precise upper bound on the function sinp(x) +cosp(x), where sinp(x) denotes parabolic sine and cosp(x) parabolic cosine.
Let B(H) denote the C-& lowast;-algebra of all bounded linear operators on a complex separable Hilbert space H, and let |||||| be a unitarily invariant norm on some ideal of B(H). In this paper, we shall show that: for f, g be two continuous non-negative real-valued functions defined on the non-negative half-line [0,infinity[ satisfying f (t)g(t) = t, for all t >= 0, and for two positive operators P,Q in B(H), the following inequality holds: for all X is an element of , |||f(P)Xg(Q)+g(P)Xf(Q)|||>= 2|||P-1/(2)XQ(1)/(2)|||.
We consider univalent functions, analytic in the unit disc |z| < 1 in the complex plane C which map |z| <1 onto a domain with some property. In M. Nunokawa, J. Soko & lstrok;, E. Trybucka, On some sufficient conditions for a function to be p-valent starlike, Symmetry, 11 (11) (2019), Art. 1417, we have established a sufficient condition for a function analytic in |z| < 1 to be p-valent starlike in |z| < 1. In this paper we shall determine the new sufficient conditions for the starlikeness of order alpha for p-valent functions and some symmetric results.
The aim of the present paper is to show the best possible bound of quadratic fractional type for one-term refinements and reverses of the Young inequality involving the Kantorovich constant. The key tool for obtaining the results is a new L' Hopital-type higher monotone rule. Some applications to operator inequalities and to the theory of matrices are also discussed.
In this work, we first construct a non-monotonic discrete kernel function in the whole plane, where the parameters in the newly constructed kernel function are restricted to two special subset of the real line. Utilizing some classic techniques from real analysis and converting the weight functions in the whole plane to the first quadrant, we established the estimation formula for the weight functions, and then a class of new Hilbert-type inequality and its equivalent forms are established. In addition, it is proved that the constant factor of the newly obtained inequality is optimal. Moreover, assigning some special values to the parameters in the kernel function, some special inequalities are proved at the end of the paper.
This paper investigates novel refinements and reversals of Jensen's inequality within the framework of generalized convexity, particularly focusing on MW-convex functions. These functions extend classical convexity by incorporating nonlinear mean-type structures via a strictly monotonic function W. We present sharper forms of both Jensen and Jensen-Mercer inequalities, providing double-sided bounds and reverse inequalities that significantly improve classical results. Building upon recent advances, our contributions include enhanced inequalities adapted to k-harmonically and k-geometrically convex functions. These extensions are achieved by selecting specific transformation functions W, such as W(t) = 1/(t-k) and W(t) = log(t-k), which yield new insights into the structure of generalized convexity. Furthermore, we establish Jensen-type inequalities in operator settings, leveraging harmonic convexity. Our operator inequalities yield refined spectral bounds and deepen the connection between convexity and functional analysis. In particular, a new operator-Jensen inequality and a McCarthy-type inequality are proved for the class of harmonic convex functions. Altogether, this unified treatment of generalized convexity broadens the applicability of classical inequalities and offers powerful tools for future studies in analysis, optimization, information theory, and operator theory.
In this paper, the closure of analytic type tent spaces within the weighted-type space is characterized. The boundedness and compactness of generalized weighted composition operators on the closure of analytic type tent spaces in the weighted-type space are studied. Additionally, the boundedness, compactness, and essential norm of generalized weighted composition operators from the weighted-type space to analytic type tent spaces are also investigated.
We study the concepts of Birkhoff-James orthogonality and parallelism in Hilbert space operators, induced by the operator radius norm w rho (center dot). In particular, we completely characterize Birkhoff-James orthogonality and parallelism with respect to w rho (center dot). As an application of the results presented, we obtain a well-known characterization due to R. Bhatia and P. Semrl for the classical Birkhoff-James orthogonality of Hilbert space operators. Some other related results are also discussed.
Let Bnp ( p 1) be the unit ball of lnp and Gamma m(Bnp) be the smallest positive number gamma such that Bnp can be covered by m translates of gamma Bnp . By using different configurations of translates of gamma Bnp , we obtain a universal upper bound of Gamma 2n (Bnp) for fixed p E [1, infinity], a nontrivial upper bound for Gamma 2n (Bnp) for all p E [1, infinity] when n is small, and a useful upper bound of Gamma 2n (Bnp) when n and p are both large. It is still not clear whether there exists a constant c E (0,1) such that Gamma 2n (Bnp) c holds whenever p 1 and n 2.
Let W-1,W-n(R-n) (n >= 2) be the standard Sobolev space, and denote, for p>n gamma 1=infu is an element of W1,n(Rn),u equivalent to 0 integral Rn(Fn(del u)+|u|n)dx(integral Rn|u|pdx)np where F: R -> [0,0 infinity) be a convex function of class C-2 (R-n\{0}), which is even and positively homogeneous of degree 1. For gamma epsilon (0,gamma(1)), we define a norm in W-1,W-n(R-n) by & Vert;u & Vert;(F,n,gamma,p)=(integral(Rn)(F-n(del u)+|u|(n))dx-gamma(integral(Rn)|u|(p)dx)(n/p))(1/n). By performing a blow-up analysis, we prove that for real numbers 0 <= gamma < gamma(1)and p > n, the following anisotropic Trudinger-Moser inequality Sup u is an element of W1,n(Rn),& Vert;u & Vert;F,n,gamma,p <= 1 integral(Rn)Phi(lambda(n)|u| n/n-1)dx can be attained by some function u(0)is an element of W-1,W-n(R-n)with & Vert;u(0)& Vert;(F,n,gamma,p)=1, where Phi(t)=e(t)-& sum;(n-1)(j=0)tj/j!,lambda n=n/n(n-1)kappa 1/n-1 n and kappa(n) is the volume of the unit Wulff ball. In the case gamma=0, this is reduced to a result of Zhou-Zhou [19]
We characterize the pairs of weights (u,v) for the maximal operator G0 , defined for nonnegative functions on (0, infinity) by G(0)f(x)=sup(b>x)exp(1/b integral(b)(0)logf), to be bounded from L-p(v) to L-q(u), p <= q, or from L-p(v) to L-q,L-infinity(u).
In this paper, we use alpha-Bernstein operators to solve an open problem related to box-convex functions. Future targeted applications reside in the area of stochastic optimization in problems such as AUC maximization and stochastic programs with chance constraints.
We provide a sufficient condition for expanding the domain in the Schrodinger-Robertson uncertainty inequality for infinitesimal operators derived from a unitary representation of a Lie group.
We present a new formulation of the Hilbert transform constructed via the q-deformation of convolution, which is called the q-deformed Hilbert transform. We also find the q-deformed Hilbert transform of some basic functions and examine its connection with the q-Fourier transform. In particular, a number of new related inequalities and embeddings are proved such as a q-analuge of the Chebyshev inequality and a Hardy-type inequality. In additionally, we present a direct application of Hardy-type inequality to study some inequalities for the qdeformed Hilbert transform on Lp(Rq) and Lp,r(Rq). Finally, we prove a weak (1.1) inequality for the q-deformed Hilbert transform.
In this article, we study some inequalities for the h-Fourier cosine-Laplace discrete generalized convolution on the time scale T0 h and establish some norm estimations for this discrete generalized convolution on some function spaces. We present some sufficient conditions for the existence of the h-Fourier cosine-Laplace discrete generalized convolution. A Young-type inequality, a Saitoh-type inequality and a reverse Saitoh-type inequality for this discrete generalized convolution are obtained. As applications, we apply some of these inequalities to estimate the solutions of a class of equations of the h-Fourier cosine-Laplace discrete generalized convolution type.
We provide a new approach to the proofs of some existing bilateral inequalities for fusion frames in Hilbert spaces from the perspective of function theory, which greatly simplifies the proving process and shows that the parameter involving in four of the results mentioned can take values from larger ranges. We also present an improvement to two results on this topic. At the end of the paper we establish several new bilateral inequalities for fusion frames in Hilbert spaces, following the approaches of which corresponding bilateral inequalities for some other generalized frames with new types of structures can be naturally obtained.
The study of inequalities in various norms for polynomials and their derivatives in the plane is fundamental to geometric function theory. This paper focuses on Zygmund-type norm estimates for polynomials that do not vanish in a positive-radius disc. We establish integral norm estimates for the growth of higher-order derivatives of a polynomial in the plane, including extensions of several important inequalities of approximation theory and related inequalities established by Jain [Turk. J. Math. 31 (2007), 89-94].
In this paper, we establish bounds and prove the complete monotonicity of functions involving the Nielsen beta function. We derive a novel integral representation for the Dirichlet eta function. As an application, we construct several bounded Bernstein functions associated with the Nielsen beta function.
This paper extends the boundedness of the one-sided maximal operators from the Lebesgue spaces with variable exponents to the one-sided Herz spaces with variable exponents. The main result generalizes the boundedness of the one-sided maximal operators on the classical Herz spaces.
