Mozyrska-Torres logarithm composed with forward jump

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Theorem

Let $\mathbb{T}$ be a time scale. Then, $$L_{\mathbb{T}}(\sigma(t)) = L_{\mathbb{T}}(t) + \dfrac{\mu(t)}{t},$$ where $L_{\mathbb{T}}$ denotes the Mozyrska-Torres logarithm, $\sigma$ denotes the forward jump, and $\mu$ denotes the forward graininess.

Proof

References

Dorota Mozyrska and Delfim F. M. Torres: The Natural Logarithm on Time Scales (2008)... (previous)... (next)