δ ( n , k ) a k ζ ( β + ( 1 γ μ ) ) (11)

This establishes our proof.

Corollary 2.1.

If f B λ n ( μ , β , ζ ) then

a k ζ ( β + ( 1 γ μ ) ) k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) , k = 2 , 3 , (12)

equality is attained for

f ( z ) = z + ζ ( β + ( 1 γ μ ) ) k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) z k , k = 2 , 3 , (13)

We shall state the growth and distortion theorems for the class B λ n ( ϑ , ς , γ , ι ) The results of which follow easily on applying Theorem 2.1, therefore, we deem it necessary to omit the trivial proofs.

2.2. Growth and Distortion Theorems

Theorem 2.2.

Let the function f ( z ) B λ n ( μ , β , ζ , γ ) then for | z | = r

r ζ ( β + ( 1 γ μ ) ) 2 ( 1 + ζ β ) [ 1 + λ ] n δ ( n , 2 ) r 2 | f ( z ) | r + ζ ( β + ( 1 γ μ ) ) 2 ( 1 + ζ β ) [ 1 + λ ] n δ ( n , 2 ) r 2

Theorem 2.3.

Let the function f ( z ) B λ n ( μ , β , ζ ) then for | z | = r

1 ζ ( β + ( 1 γ μ ) ) ( 1 + ζ β ) [ 1 + λ ] n δ ( n , 2 ) r | f ( z ) | 1 + ζ ( β + ( 1 γ μ ) ) ( 1 + ζ β ) [ 1 + λ ] n δ ( n , 2 ) r

When f ( z ) = z + ζ ( β + ( 1 γ μ ) ) 2 ( 1 + ζ β ) [ 1 + λ ] n δ ( n , 2 ) z 2

we obtain a sharp result.

2.3. Radii of Close-to-Convexity, Starlikeness and Convexity

Theorem 2.4.

Let the function f ( z ) B λ n ( μ , β , ζ , γ ) , then f is close-to-convex of order δ in | z | < R τ 1

where

R τ 1 = inf k 2 [ k ( 1 + ζ β ) ( 1 δ ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) ζ ( β + ( 1 γ μ ) ) ] 1 k 1

The result obtained is sharp.

Proof.

It is sufficient to show that | f ( z ) 1 | 1 δ for | z | < R τ Thus we can write

| f ( z ) 1 | = | n = 2 k a k z k 1 | n = 2 k a k | z | k 1

Therefore | f ( z ) 1 | 1 δ if

n = 2 ( k 1 δ ) a k | z | k 1 1 (14)

But we have from theorem 2.1. that

k = 2 k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) a k ζ ( β + ( 1 γ μ ) ) 1 (15)

Relating (14) and (15) we have our desired result.

Theorem.2.5.

Let the function f ( z ) B λ n ( μ , β , ζ ) , then f is starlike of order δ , 0 δ < 1 in | z | < R τ 2

where

R τ 2 = inf k 2 [ k ( 1 + ζ β ) ( 1 δ ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) ( k δ ) ζ ( β + ( 1 γ μ ) ) ] 1 k 1

The result obtain here is sharp.

Proof.

We must show that | z f ( z ) f ( z ) 1 | 1 δ for | z | < R τ 2 . Equivalently, we have

k = 2 ( k δ ) a k | z k 1 | 1 δ 1 (16)

But we have from theorem 2.1. that

k = 2 k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) a k ζ ( β + ( 1 γ μ ) ) 1 (17)

Relating (16) and (17) will have our desired result.

Theorem 2.6.

Let the function f ( z ) B λ n ( ϑ , ς , γ , ι ) , then f is convex of order δ , 0 δ < 1 in | z | < R τ 3

where

R τ 3 = inf k 2 [ ( 1 + ζ β ) ( 1 δ ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) ( k δ ) ζ ( β + ( 1 γ μ ) ) ] 1 k 1

The result obtain here is sharp.

Proof.

By using the technique of theorem 2.5 we easily show that | z f ( z ) f ( z ) | 1 δ this holds for | z | < R τ 3 . The analogous details of theorem 2.5 are thus omitted, hence the proof.

3. Integral Operator

Theorem 3.1.

Let the function f ( z ) defined by (2) be in the class T n ( μ , β , γ ζ ) and let c be a real number such that c > 1 . Then the function defined by

F ( z ) = c + 1 z c t c 1 f ( t ) d t (18)

also belong to the class T n ( μ , β , γ , ζ )

Proof.

From the representation and definition of F ( z ) we have that

F ( z ) = z k = 2 b k z k (19)

where

b k = ( c + 1 c + k ) a k (20)

Thus we have

k = 2 k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) b k (21)

= k = 2 k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) ( c + 1 c + k ) a k

k = 2 k ( 1 + ζ β ) [ 1 + ( k 1 ) λ ] n δ ( n , k ) a k ζ ( β + ( 1 γ μ ) ) (22)

since f ( z ) T n ( μ , β , γ , ζ ) . By theorem 1.1 F ( z ) T n ( μ , β , γ , ζ ) . This establishes our proof.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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https://doi.org/10.1090/S0002-9939-1975-0367176-1
[4] Al-Oboudi, F.M. (2004) On Univalent Functions Defined by a Generalized Salagean Operator. Indian Journal of Pure and Applied Mathematics, 25-28, 1429-1436.
https://doi.org/10.1155/S0161171204108090
[5] Lupas, A.A. (2011) Certain Differential Superordination Using a Generalized Salagean and Ruscheweyh Operator. Acta Universitatis Apulensis, 25, 31-40.
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https://doi.org/10.12988/imf.2007.07253

  
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