Abstract
We first define the q-analogue operators of fractional calculus which are then used in defining
certain classes of functions analytic in the open disk. The results investigated for these classes of
functions include the coefficient inequalities and some distortion theorems. The results provide
extensions of various known results in the q-theory of analytic functions. Special cases of our
results are pointed out briefly.
Results & Lemmas (14)
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Proposition 1.
Proposition 1. Let α > 0 and λ > −1. Then (2.7) I α q,zzλ = q(λ + 1) q(λ + α + 1)zα+λ.
Proposition 1. Let α > 0 and λ > −1. Then (2.7) I α q,zzλ = q(λ + 1) q(λ + α + 1)zα+λ.
Proposition 2.
Proposition 2. Let α ≥0 and λ > −1. Then (2.9) Dα q,zzλ = q(λ + 1) q(λ −α + 1)zλ−α. We omit here the proof of Proposition 2 as its…
Proposition 2. Let α ≥0 and λ > −1. Then (2.9) Dα q,zzλ = q(λ + 1) q(λ −α + 1)zλ−α. We omit here the proof of Proposition 2 as its details are similar to Propos- ition 1. Our aim in this paper is to introduce some new classes of functions defined by using fractional q-calculus operators which are analytic in the open disk. We also derive some results giving various coefficient inequalities and distortion theorems involving the fractional q-calculus operators. Special cases of the results are als
Theorem 1.
Theorem 1. A function f of the form (3.2) belongs to the class J α q,δ if and only if (3.8) ∞ k=n+1 q(k + 1)q(2 −α) q(2)q(k −α + 1)(1…
Theorem 1. A function f of the form (3.2) belongs to the class J α q,δ if and only if (3.8) ∞ k=n+1 q(k + 1)q(2 −α) q(2)q(k −α + 1)(1 + β)ak ≤2β(1 −δ). The result is sharp.
Theorem 2.
Theorem 2. A function f of the form (3.2) belongs to the class S α q,σ if and only if (3.12) ∞ k=n+1 q(k + 1)q(2 −α) q(2)q(k −α +…
Theorem 2. A function f of the form (3.2) belongs to the class S α q,σ if and only if (3.12) ∞ k=n+1 q(k + 1)q(2 −α) q(2)q(k −α + 1)ak (1 −σ)(1 −q) + σ(1 −qk−α) ≤(1 −β −σ)(1 −q) + σ(1 −q1−α). The result is sharp.
Theorem 3.
Theorem 3. Let the function f be defined by (3.1) be in the class α q. Then (3.17) ∞ k=n+1 q(k + 1) q(k −α + 1)|ak| ≤(1 −β)q(2) q(2…
Theorem 3. Let the function f be defined by (3.1) be in the class α q. Then (3.17) ∞ k=n+1 q(k + 1) q(k −α + 1)|ak| ≤(1 −β)q(2) q(2 −α) . The result is sharp.
Theorem 4.
Theorem 4. Let the function f (z) defined by (3.2) be in the class J α q,δ (−∞< α < 2, 0 < q < 1). Then (4.1) |z| −2β 1 −δ 1 + β B(n, α,…
Theorem 4. Let the function f (z) defined by (3.2) be in the class J α q,δ (−∞< α < 2, 0 < q < 1). Then (4.1) |z| −2β 1 −δ 1 + β B(n, α, q)|z|n+1 ≤|f (z)| ≤|z| + 2β 1 −δ 1 + β B(n, α, q)|z|n+1 (z ∈U),
Corollary 1.
Corollary 1. Let the function f (z) defined by (3.2) be in the class J α q,δ. Then (4.7) q(2) q(2 −α)|z|1−α 1 −2β 1 −δ 1 + β |z|n
Corollary 1. Let the function f (z) defined by (3.2) be in the class J α q,δ. Then (4.7) q(2) q(2 −α)|z|1−α 1 −2β 1 −δ 1 + β |z|n
Corollary 2.
Corollary 2. Let the function f (z) be in the class J α q,δ. Then (4.8) q(2) q(2 + α)|z|1+α 1 −2β 1 −δ 1 + β |z|n
Corollary 2. Let the function f (z) be in the class J α q,δ. Then (4.8) q(2) q(2 + α)|z|1+α 1 −2β 1 −δ 1 + β |z|n
Theorem 5.
Theorem 5. Let the function f (z) defined by (3.2) be in the class S α q,σ (−∞< α < 2, 0 < q < 1). Then for z ∈U: (4.9) |z| −B(n, α,…
Theorem 5. Let the function f (z) defined by (3.2) be in the class S α q,σ (−∞< α < 2, 0 < q < 1). Then for z ∈U: (4.9) |z| −B(n, α, q)C|z|n+1 ≤|f (z)| ≤|z| + B(n, α, q)C|z|n+1. Also (4.10) |z|−CD|z|n+1 ≤|z λ q,zf (z)| ≤|z|+CD|z|n+1 (−∞< λ < 2),
Corollary 3.
Corollary 3. Let 0 ≤λ < 2, n ∈N and the function f (z) defined by (3.2) be in the class S α q,σ. Then for all z ∈U: (4.17) q(2) q(2…
Corollary 3. Let 0 ≤λ < 2, n ∈N and the function f (z) defined by (3.2) be in the class S α q,σ. Then for all z ∈U: (4.17) q(2) q(2 −λ)|z|1−λ{1 −CD|z|n} ≤|Dλ q,zf (z)| ≤ q(2) q(2 −λ)|z|1−λ{1 + CD|z|n}. and
Corollary 4.
Corollary 4. Let λ > 0, n ∈N and the function f (z) be in the class S α q,σ. Then for all z ∈U: (4.18) q(2) q(2 + λ)|z|1+λ 1 −CD|z|n ≤|I…
Corollary 4. Let λ > 0, n ∈N and the function f (z) be in the class S α q,σ. Then for all z ∈U: (4.18) q(2) q(2 + λ)|z|1+λ{1 −CD|z|n} ≤|I λ q,zf (z)| ≤ q(2) q(2 + λ)|z|1+λ{1 + CD|z|n}, where C is the expression given by (4.11), and D is the expression given by (4.12) (with λ replaced by −λ therein).
Theorem 6.
Theorem 6. Let the function f (z) defined by (3.1) be in the class α q (−∞< α < 2, 0 < q < 1). Then for z ∈U: (4.19) |z| −(1 −β)B(n, α,…
Theorem 6. Let the function f (z) defined by (3.1) be in the class α q (−∞< α < 2, 0 < q < 1). Then for z ∈U: (4.19) |z| −(1 −β)B(n, α, q)|z|n+1 ≤|f (z)| ≤|z| + (1 −β)B(n, α, q)|z|n+1, where B(n, α, q) is given by (4.2). Furthermore (4.20) |z| −(1 −β)D|z|n+1 ≤|z λ q,zf (z)| ≤|z| + (1 −β)D|z|n+1 (−∞< λ < 2), where D is given by (4.12). The proof of the above distortion theorem is similar to Theorem 5, details
Corollary 5.
Corollary 5. If 0 ≤λ < 2, n ∈N and the function f (z) defined by (3.1) is in the class α q, then (4.21) q(2) q(2 −λ)|z|1−λ 1 −(1 −β)D|z|n…
Corollary 5. If 0 ≤λ < 2, n ∈N and the function f (z) defined by (3.1) is in the class α q, then (4.21) q(2) q(2 −λ)|z|1−λ{1 −(1 −β)D|z|n} ≤|Dλ q,zf (z)| ≤ q(2) q(2 −λ)|z|1−λ{1 + (1 −β)D|z|n}. and
Corollary 6.
Corollary 6. If λ > 0, n ∈N and the function f (z) is in the class α q, then (4.22) q(2) q(2 + λ)|z|1+λ 1 −(1 −β)D|z|n ≤|I λ q,zf (z)| ≤…
Corollary 6. If λ > 0, n ∈N and the function f (z) is in the class α q, then (4.22) q(2) q(2 + λ)|z|1+λ{1 −(1 −β)D|z|n} ≤|I λ q,zf (z)| ≤ q(2) q(2 + λ)|z|1+λ{1 + (1 −β)D|z|n}, for all z ∈U, where D is the expression given by (4.12) (with λ replaced by −λ therein). 5. Concluding observations and remarks We briefly consider now some consequences of the results derived in the pre- ceeding sections. If we let q →1−, and make use of the limit formula (1.10),