Ma-Minda φ-classes studied in this paper:
Abstract
In this article, we studied the geometric properties of generalized Wright-Bessel
functions.
For this purpose, we determined sufficient conditions for univalency, convexity,
starlikeness and close-to-convexity of the generalized Wright-Bessel functions in the open unit
disk.
2020 Mathematics Subject Classification. 33C10, 30C45, 30C55.
Key words and phrases. Generalized Wright-Bessel function; univalent; starlike; convex;
close-to-convex.
Results & Lemmas (11)
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Lemma 1.1
Lemma 1.1 ([8]). Let f define by (1) and suppose that 1 ≥2a2 ≥· · · ≥kak ≥· · · ≥0 or 1 ≤2a2 ≤· · · ≤kak ≤· · · ≤2. Then f is regular and…
Lemma 1.1 ([8]). Let f define by (1) and suppose that 1 ≥2a2 ≥· · · ≥kak ≥· · · ≥0 or 1 ≤2a2 ≤· · · ≤kak ≤· · · ≤2. Then f is regular and univalent in U. Following the proof of Ozaki it can be proved that if a function f satisfies the conditions given in Lemma 1.1, then f is close-to-convex with respect to the convex function −log(1 −z).
Lemma 1.2
Lemma 1.2 ([9]). If the function f ∈A, satisfy |(f(z)/z) −1| < 1 for each z ∈U, then f is univalent and starlike in U1/2 = z: |z| < 1/2.
Lemma 1.2 ([9]). If the function f ∈A, satisfy |(f(z)/z) −1| < 1 for each z ∈U, then f is univalent and starlike in U1/2 = {z : |z| < 1/2}.
Lemma 1.3
Lemma 1.3 ([10]). If the function f ∈A, satisfy |f ′(z)−1| < 1 for each z ∈U, then f is convex in U1/2.
Lemma 1.3 ([10]). If the function f ∈A, satisfy |f ′(z)−1| < 1 for each z ∈U, then f is convex in U1/2.
Lemma 1.4
Lemma 1.4 ([11]). Assume that f ∈A. Then the following results hold true: (i) If zf ′(z) f(z) −1 < 1 2, then f ∈Sp. (ii) If zf ′′(z) f ′(z)…
Lemma 1.4 ([11]). Assume that f ∈A. Then the following results hold true: (i) If zf ′(z) f(z) −1 < 1 2, then f ∈Sp. (ii) If zf ′′(z) f ′(z) < 1 2, then f ∈UCV . 2. Main results
Theorem 2.1.
Theorem 2.1. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462, then Jµ v,λ(z) given in (2) is close-to-convex with respect to convex function…
Theorem 2.1. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462, then Jµ v,λ(z) given in (2) is close-to-convex with respect to convex function −log(1−z) and hence univalent in U.
Theorem 2.2.
Theorem 2.2. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 2 + √ 2 4 where [|µ|] denotes the greatest integer…
Theorem 2.2. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 2 + √ 2 4 where [|µ|] denotes the greatest integer value of µ, then Jµ v,λ(z) given in (2) is starlike in U.
Theorem 2.3.
Theorem 2.3. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 2
Theorem 2.3. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 2
Theorem 2.4.
Theorem 2.4. Let v, µ > 0 and λ > −1/2. If λ + v ≥0, 462, then Jµ v,λ(z) given in (2) is univalent and starlike in U1/2.
Theorem 2.4. Let v, µ > 0 and λ > −1/2. If λ + v ≥0, 462, then Jµ v,λ(z) given in (2) is univalent and starlike in U1/2.
Theorem 2.5.
Theorem 2.5. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 2 + √ 2 4 where [|µ|] denotes the greatest integer…
Theorem 2.5. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 2 + √ 2 4 where [|µ|] denotes the greatest integer value of µ, then Jµ v,λ(z) given in (2) is convex in U1/2.
Theorem 2.6.
Theorem 2.6. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 1 8(5 + √ 17) where [|µ|] denotes the greatest…
Theorem 2.6. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 1 8(5 + √ 17) where [|µ|] denotes the greatest integer value of µ, then the function Jµ v,λ(z) given in (2) is belongs to the class of Sp.
Theorem 2.7.
Theorem 2.7. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 3 where [|µ|] denotes the greatest integer value of…
Theorem 2.7. Let v, µ > 0 and λ ≥−1/2. If λ + v ≥0, 462 and (λ + 1)(v + λ + 1)[|µ|] > 3 where [|µ|] denotes the greatest integer value of µ, then the function Jµ v,λ(z) given in (2) is belongs to the class of UCV . Example 2.7. The function J2 2,2(z) is in the class UCV . References [1] R.S. Pathak, Certain convergence theorems and asymptotic properties of a generalization of Lommel and Maitland transformations, Proc. Natl. Acad. Sci. India A 36 (1966), no. 1, 81–86. [2] V. Kiryakova, A Guide to
Function classes studied:
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