Sentence examples for obtained some improved from inspiring English sources
For instance, Hu and Xue [5] obtained some improved reverses of Young type inequalities for matrices, Zou and Peng [6] presented some trace inequalities for matrix means, Zou and Jiang [7] gave a note on interpolation between Cauchy-Schwarz matrix norm inequalities and the arithmetic-geometric mean.
In this paper, we obtain some improved reverses of Young type inequalities which were established by Burqan and Khandaqji (J. Math. Inequal. 9 113-120, 2015).
In this work, we further study the strong convergence for weighted sums of arrays of ANA random variables without assumption of identical distribution and obtain some improved results (i.e., the so-called complete moment convergence, which will be introduced in Definition 1.5).
For Equation (2), Belaïdi [4], Belaïdi and El Farissi [5] investigated the fixed points and the relationship between small functions and differential polynomials of solutions of Equation (2) and obtained some results which improve Theorems D and C.
and obtained some oscillation criteria, which improved and extended the results that were established in [19 21].
The authors obtained some new oscillation criteria which improved the results established by Zhang and Zhu [10] and Şahiner [11].
Besides, by applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, Zhang and Li [28] investigated the existence and multiplicity of fast homoclinic solutions for a class of nonlinear second-order nonautonomous systems and obtained some results, which generalized and improved problem (1.4).
In [18], Sun and Saker considered equation (1.5) and obtained some new oscillation criteria, which improve and complement that given in [16].
Some applications of the obtained results improve some existing results on neural networks.
The phases obtained were improved by using autoSHARP [18].
The mitogenomes obtained allowed improved resolution of the mtDNA phylogeny within these Native American haplogroups.
