  We prove a few novel state-dependent uncertainty relations for product as
well the sum of variances of two incompatible observables. These uncertainty
relations are shown to be tighter than the Roberson-Schr\"odinger uncertainty
relation and other ones existing in the current literature. Also, we derive
state dependent upper bound to the sum and the product of variances using the
reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our
results suggest that not only we cannot prepare quantum states for which two
incompatible observables can have sharp values, but also we have both, lower
and upper limits on the variances of quantum mechanical observables at a
fundamental level.
