a note on sharpening of erdös-lax and turán-type inequalities for a constrained polynomial

The well-known erdös-lax and turán-type inequalities, which relate the uniform norm of a univariate complex coefficient polynomial to its derivative on the unit circle in the plane, are discussed in this paper. we create some new inequalities here when there is a restriction on its zeros. the obtained results strengthen some recently proved erdös-lax and turán-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in a very rich literature on this subject.