In this study, we introduced several novel Hardy-type inequalities with negative parameters for monotone functions within the framework of delta calculus on time scales $ \mathbb{T} $. As an application, when $ \mathbb{T = N}_{0}, $ we derived discrete inequalities with negative parameters for monotone sequences, offering fundamentally new results. When $ \mathbb{T = R}, $ we established continuous analogues of inequalities that have appeared in previous literature. Additionally, we presented inequalities for other time scales, such as $ \mathbb{T} = q^{\mathbb{N}_{0}} $ for $ q > 1, $ which, to the best of the authors' knowledge, represented largely novel contributions.
Citation: Ahmed M. Ahmed, Ahmed I. Saied, Mohammed Zakarya, Amirah Ayidh I Al-Thaqfan, Maha Ali, Haytham M. Rezk. Advanced Hardy-type inequalities with negative parameters involving monotone functions in delta calculus on time scales[J]. AIMS Mathematics, 2024, 9(11): 31926-31946. doi: 10.3934/math.20241534
In this study, we introduced several novel Hardy-type inequalities with negative parameters for monotone functions within the framework of delta calculus on time scales $ \mathbb{T} $. As an application, when $ \mathbb{T = N}_{0}, $ we derived discrete inequalities with negative parameters for monotone sequences, offering fundamentally new results. When $ \mathbb{T = R}, $ we established continuous analogues of inequalities that have appeared in previous literature. Additionally, we presented inequalities for other time scales, such as $ \mathbb{T} = q^{\mathbb{N}_{0}} $ for $ q > 1, $ which, to the best of the authors' knowledge, represented largely novel contributions.
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Ahmed M. Ahmed, Ahmed I. Saied, Mohammed Zakarya, Amirah Ayidh I Al-Thaqfan, Maha Ali, Haytham M. Rezk. Advanced Hardy-type inequalities with negative parameters involving monotone functions in delta calculus on time scales[J]. AIMS Mathematics, 2024, 9(11): 31926-31946. doi: 10.3934/math.20241534
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