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In this paper, we obtain an improved upper bound involving the systole andarea for the volume entropy of a Riemannian surface. As a result, we show thatevery orientable and closed Riemannian surface of genus $g\geq 18$ satisfiesLoewner's systolic ratio inequality. We also show that every closed orientableand nonpositively curved Riemannnian surface of genus $g\geq 11$ satisfiesLoewner's systolic ratio inequality.

Every closed surface of genus at least 18 is Loewner