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.
