Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta function

Abstract

Applying techniques of real analysis and weight functions, we study some equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular nonhomogeneous kernel. The constant factors are related to the Riemann zeta function and are proved to be best possible. In the form of applications, we deduce a few equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular homogeneous kernel. We also consider some corollaries as particular cases.

On Hardy-type integral inequalities in the whole plane related to the extended Hurwitz-zeta function

“…When statement (iii) is satisfied, if there exists a constant M 1 ≤ K (1) (σ ) such that (15) is valid, then by the proof of Lemma 1, we have K (1) (σ ) ≤ M 1 . It follows that the constant factor M 1 = K (1) (σ ) in (15) is the best possible.…”

“…The constant factor M 1 = K (1) (σ ) in (14) is still the best possible. Otherwise, by (16) (for σ 1 = σ ), we would conclude that the constant factor M 1 = K (1) (σ ) in (15) was not the best possible.…”

“…Furthermore, Hong [10] considered as well an equivalent condition between a Hilbert-type inequality with homogenous kernel and a few parameters. Some additional kinds of Hilbert-type inequalities were also obtained in [11][12][13][14][15][16][17][18][19]. Most of these results are constructed in the quarter plane of the first quadrant.…”

Equivalent Parameter Conditions for the Validity of Half-Discrete Hilbert-Type Multiple Integral Inequality with Generalized Homogeneous Kernel

“…Finally, the boundedness and norm of the corresponding series operator and integral operator are discussed. The relevant literature can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”

On The Third-Order Complex Differential Inequalities of ξ-Generalized-Hurwitz–Lerch Zeta Functions

“…This theory is developed with numerous implementations and generalizations by various researchers. One may refer to Al-Janaby et al [42,43], Ghanim [44], Ghanim and Darus [45], Ghanim and Al-Janaby [46], Rǎducanu and Srivastava [47], Srivastava and Attiya [48], Srivastava et al [49,50], Xing and Jose [51], Choi and Srivastava [52], Milovanovic and Rassias [53] and Rassias and Yang [54][55][56][57].…”