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Some inequalities associated with a linear operator defined for a class of multivalent functions. (English) Zbl 1054.30013

Using the notation (λ) k (for any λ0) defined by

(λ) k =λ(λ+1)(λ+k-1)(k=1,2,)

define

ϕ p (a,c,z)=z p + k=1 (a) k (c) k z k+p (|z|<1,a,c,c0,-1,-2,)·

Using ϕ p and the Hadamard convolution * define

L p (a,c)f(z):=ϕ p (a,c,z)*f(z),

where

f(z)=z p + k=p+n a k z k (1)

is an analytic function in the open unit disc U of the complex plane. We shall denote by A(p,n;a,c,α) the class of all analytic functions f(z) of the form (1) satisfying

ReL p (a+2,c)f L p (a+1,c)f<α(zU,α>1,a,c,c0,-1,-2,)·

Let D δ+p-1 f denote z (1-z) δ+p *f(z). The authors derive various inequalities involving L p (b,c,f) and D λ f for functions in A(p,n;a,c,α) for various parameters b and λ. They also obtain results involving differential subordination between analytic functions in their class of functions.

MSC:
30C45Special classes of univalent and multivalent functions