Since is continuous on the interval while , so by the 's inequality, we have
In fact, this inequality is a side of the so-called 's inequality:
's inequality.
Let be continuous on the interval , and Then holds
Formerly we've proved the left side, now we try to illustrate the right side.
Note that
i.e.
Integrate by two sides, we get
From inequality, we know
Thus
Actually, we could even generalize the 's inequality to
Generalized 's inequality.
Let be continuous on the interval , and while is a real constant. Then there holds
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