Normalized generalized Bessel function and its geometric properties

Lemma 1.1 ([8]) The following inequality holds for t ∈(0,∞): lnt – 1 t < ψ(t) < lnt – 1 2t , (1.5) where ψ represents the digamma function, that is the derivative of the logarithm of  func- tion.

Lemma 1.2 ([6, Satz IX]) If {Aj}n∈N is a nonnegative real sequence with A1 = 1, such that {jAj}j∈N and {jAj – (j + 1)Aj+1}j∈N are nonincreasing sequences, then f (z) = ∞ j=1 Ajzj is star- like in U.

Lemma 1.3 ([23, Corollary 7 and Theorem 8′]) Assume that 0 ≤jAj ≤··· ≤2A2 ≤1, or 2 ≥jAj ≥··· ≥2A2 ≥1, where f can be expressed by f (z) = z + ∞ j=2 Ajzj, z ∈U, then the function f is close-to-convex with respect to –log(1 – z).

Lemma 1.4 Assume that f (z) = z + ∞ j=1 A2j+1z2j+1, z ∈U, is an odd function such that 0 ≤(1 + 2j)A2j+1 ≤··· ≤3A3 ≤1, or 2 ≥(1 + 2j)A2j+1 ≥··· ≥3A3 ≥1, for all n ∈N. Then, f ∈Cg∗⊂S. 2 Main results The first two theorems of this section contain some interesting and useful results involving the order of starlikeness and the order of convexity of Uσ,r. The proofs use the inequalities for the digamma function and its derivative that have been proved in [8].

Theorem 2.1 Let σ ∈(–1,0) ∪(0,+∞) and r ∈C, such that ln(1 + σ) + ln2 – 1 1 + σ – 3 2 – ln|r| ≥0, (2.1) and 0 ≤α ≤3|σ| – 2|r| 3|σ| – |r| . (2.2) Then, Uσ,r ∈S∗(α).

Theorem 2.2 Let σ ∈(–1,0) ∪(0,+∞) and r ∈C, such that ln(σ + 1) + ln2 – 1 σ + 1 – 2 – ln|r| ≥0, (2.10) and 0 ≤α ≤3|σ| – 4|r| 3|σ| – 2|r|. (2.11) Then, Uσ,r ∈K(α).

Theorem 2.3 Let σ ≥r with r ∈(0,+∞). Then, the function z 1+z ∗Uσ,r(z) is starlike in U.

Theorem 2.4 Let σ ≥r 2 with r ∈(0,+∞) and 32σ 2 + 32(1 – r)σ – 32r + 3r2 ≥0, (2.20) Then, z 1+z ∗Uσ,r(z) is starlike in U.

Theorem 2.5 Let σ ≥2r with r ∈(0,+∞). Then, the function z 1+z ∗Uσ,r(z) is convex in U.

Theorem 2.6 Let σ ≥r with r ∈(0,+∞), and suppose that 32σ 2 + 32(1 – 2r)σ + 9r2 – 64r ≥0, (2.30) Then, z 1+z ∗Uσ,r(z) is convex function in U.

Theorem 2.7 If σ ≥r 4 with r ∈(0,+∞), then (zcos√z) ∗Uσ,r(z) is a close-to-convex func- tion in U with respect to –log(1 – z).

Theorem 2.8 If σ ≥r 8 with r ∈(0,+∞), then (sinz) ∗Uσ,r(z2) z is a close-to-convex function in U with respect to log  1+z 1–z.