Let A be a unital C∗-algebra with unit e.We develop several inequalities for a positive linear functional f on A and obtain several bounds for the numerical radius v(a) of an element a∈A.Among other inequalities, we show that if ak,bk,xk∈A, r∈N and f(e)=1, then (Formula presented.) We find several equivalent conditions for v(a)=‖a‖2 and v2(a)=14‖a∗a+aa∗‖.We prove that v2(a)=14‖a∗a+aa∗‖ (resp., v(a)=‖a‖2) if and only if (Formula presented.) (resp., S12‖a‖⊆V(a)⊆D12‖a‖),where V(a) is the numerical range of a and Dk (resp., Sk) denotes the circular disk (resp., semi-circular disk) with center at the origin and radius k. We also study inequalities for the (α,β)-normal elements in A.