Higher-order q-derivatives and their applications to subclasses of multivalent Janowski type q-starlike functions

Theorem 1 If –1 ≦L < X ≦1, then S∗ (q,3)[p,v,s,X,L] ⊂S∗ (q,2)[p,v,s,X,L] ⊂S∗ (q,1)[p,v,s,X,L].

Theorem 1 gives the corresponding result that was proved by Khan et al. [8]. Thirdly, if we assign the following values to the parameters in Theorem 1: X = 1 – 2α (0 ≦α < 1) and – L = v = s + 1 = p = 1, we get the result which was proved by Wongsaijai and Sukantamala [35].

Corollary 1 (see [35]) For 0 ≦α < 1, S∗ q,3(α) ⊂S∗ q,2(α) ⊂S∗ q,1(α). Finally, below a sufficient condition for the class S∗ (q,3)[p,v,s,X,L] is given, which also includes the corresponding sufficient conditions for the function classes S∗ (q,1)[p,v,s,X,L] and S∗ (q,2)[p,v,s,X,L].

Theorem 2 A function f ∈A(p) having form (1.1) is in the class S∗ (q,3)[p,v,s,X,L] if it sat- isfies the following coefficient inequality: ∞  n=1  2ϒ(2,n) + |ϒ(3,n)|  |an+p| < |ϒ4| – 2ϒ1, (2.3)

Theorem 2 is reduced to the known result, which was stated and proved by Khan et al. [8].

Theorem 3 If –1 ≦L < X ≦1, then T S∗ (q,1)[p,v,s,X,L] ≡T S∗ (q,2)[p,v,s,X,L] ≡T S∗ (q,3)[p,v,s,X,L].

Theorem 3 is now completed. □

Theorem 3 gives the corresponding result that was proved by Khan et al. [8]. Thirdly, if we assign the following values to the parameters in Theorem 3: X = 1 – 2α (0 ≦α < 1) and – L = v = s + 1 = p = 1, we have the following known result.

Corollary 2 (see [35, Theorem 8]) If 0 ≦α < 1, then T S∗ (q,1)(α) ≡T S∗ (q,2)(α) ≡T S∗ (q,3)(α).

Corollary 3 Let the function f of the form (3.1) be in the class T S∗ q[j,p,v,s,X,L] (j = 1,2,3). Then an+p ≦ |ϒ4| – 2ϒ1 (2ϒ(2,n) + |ϒ(3,n)|). (3.4) The following function ft(z) ft(z) = zp – |ϒ4| – 2ϒ1 (2ϒ(2,1) + |ϒ(3,1)|)zp+1 (3.5) is best possible, where ϒ1, ϒ(2,n), ϒ(3,n), and ϒ4 are given by (2.4), (2.5), (2.6), and (2.7), respectively. By means of Theorem 3, it should be understood that Type 1, Type 2, and Type 3 of the

Theorem 4 If f ∈T S∗ q[j,p,v,s,X,L] (j = 1,2,3), then f (z)  ≧rp –  |ϒ4| – 2ϒ1 (2ϒ(2,1) + |ϒ(3,1)|)  rp+1 (n ∈N) zp = rp (0 < r < 1) (3.6)

Theorem 4 will give the corresponding result that was proved by Khan et al. [8]. Thirdly, if we put X = 1 – 2α (0 ≦α < 1) and – L = v = s + 1 = p = 1 in Theorem 4 and let q −→1–, we have the following known result.

Corollary 4 (see [23]) If f ∈T S∗(α), then r – 1 – α 2 – α  r2 ≦ f (z)  ≦r + 1 – α 2 – α  r2  |z| = r (0 < r < 1) 

Theorem 5 If f ∈T S∗ q[j,p,v,s,X,L] (j = 1,2,3), then f ′(z)  ≧prp–1 – (p + 1)(|ϒ4| – 2ϒ1) (2ϒ(2,1) + |ϒ(3,1)|)  rp (n ∈N) zp = rp (0 < r < 1)

Corollary 5 (see [23]) If f ∈T S∗(α), then 1 – 2(1 – α) 2 – α  r ≦ f ′(z)  ≦1 + 2(1 – α) 2 – α  r  |z| = r (0 < r < 1) 

Theorem 6 Let f ∈T S∗ q[j,p,v,s,X,L] (j = 1,2,3). Then, for |z| ≦r0(j,p,n,X,L,χ), the func- tion f is p-valent close-to-convex of order χ with (0 ≦χ < p), where r0 = inf n≥1 (2ϒ(2,n) + |ϒ(3,n)|)(p – χ) (|ϒ4| – 2ϒ1)(n + p)  1 n . (3.8) The function ft(z) given by (3.5) is best possible.

Theorem 6 gives the corresponding result that was proved by Khan et al. [8].

Theorem 7 Let f ∈T S∗ q[j,p,v,s,X,L] (j = 1,2,3). Then, for |z| ≦r1(j,p,n,X,L,χ), the func- tion f is a p-valent starlike of order χ with (0 ≦χ < p), where r1 = inf n≥1 (2ϒ(2,n) + |ϒ(3,n)|)(p – χ) (|ϒ4| – 2ϒ1)(n + p – χ)  1 n . (3.9) The result is sharp for the function ft(z) given by (3.5).

Theorem 7 gives the corresponding result that was proved by Khan et al. [8].

Corollary 6 Let f ∈T S∗ q[j,p,v,s,X,L] (j = 1,2,3). Then, for |z| ≦r2(j,p,n,X,L,χ), the function f is a p-valent convex of order χ with (0 ≦χ < p) where r2 = inf n≥1  (2ϒ(2,n) + |ϒ(3,n)|)p(p – χ) (|ϒ4| – 2ϒ1)(n + p)(n + p – χ)  1 n . (3.10) The result is sharp for the function ft(z) given by (3.5). 4 Concluding remarks and observations The works presented in this paper are basically motivated by the well-established usage of