Ma-Minda φ-classes studied in this paper:
Abstract
Sufficient conditions on associated parameters $p,b$ and $c$ are obtained so that the generalized and \textquotedblleft{normalized}\textquotedblright{} Bessel function $u_p(z)=u_{p,b,c}(z)$ satisfies $|(1+(zu''_p(z)/u'_p(z)))^2-1|<1$ or $|((zu_p(z))'/u_p(z))^2-1|<1$. We also determine the condition on these parameters so that $-(4(p+(b+1)/2)/c)u'_p(z)\prec\sqrt{1+z}$. Relations between the parameters $μ$ and $p$ are obtained such that the normalized Lommel function of first kind $h_{μ,p}(z)$ sat
Results & Lemmas (14)
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Theorem 2.1.
Theorem 2.1. Let κ, c ∈C be such that c ̸= 0 and satisfy (2.1) Re κ > max 0, |c| −3/4, then (−4κ/c)u′ p(z) ≺√1 + z. The next result gives…
Theorem 2.1. Let κ, c ∈C be such that c ̸= 0 and satisfy (2.1) Re κ > max{0, |c| −3/4}, then (−4κ/c)u′ p(z) ≺√1 + z. The next result gives sufficient conditions on the parameters κ and c so that the generalized and “normalized” Bessel function up is lemniscate convex in D.
Theorem 2.2.
Theorem 2.2. If b, p, c ∈C are such that c ̸= 0 and (2.2) √ 3|κ −2| + |c| 4 < s 9 8 + 1 √ 2, then the function up(z) is lemniscate convex…
Theorem 2.2. If b, p, c ∈C are such that c ̸= 0 and (2.2) √ 3|κ −2| + |c| 4 < s 9 8 + 1 √ 2, then the function up(z) is lemniscate convex in D. Baricz proved the recursive relation satisfied by up(z) as given in
Lemma 2.3.
Lemma 2.3. [1, Lemma 1.2, p. 14] If b, p, c ∈C and κ ̸= 0, −1, −2,..., then the function up(z) satisfies the relation 4κu′ p(z) = −cup+1(z)…
Lemma 2.3. [1, Lemma 1.2, p. 14] If b, p, c ∈C and κ ̸= 0, −1, −2, . . ., then the function up(z) satisfies the relation 4κu′ p(z) = −cup+1(z) for all z ∈C. If √ 3|κ−3|+|c|/4 < q 9 8 + 1 √ 2, then from Theroem 2.2, it follows that 1+(zu′′ p−1(z)/u′ p−1(z)) ≺√1 + z and hence (zu′
Lemma 2.3
Lemma 2.3 gives that czup(z) = −4(κ −1)zu′ p−1(z). Therefore zup(z) is lemniscate starlike in D. Thus we have the following:
Lemma 2.3 gives that czup(z) = −4(κ −1)zu′ p−1(z). Therefore zup(z) is lemniscate starlike in D. Thus we have the following:
Corollary 2.4.
Corollary 2.4. If b, p, c ∈C are such that (2.3) √ 3|κ −3| + |c| 4 < s 9 8 + 1 √ 2, then the function zup(z) is lemniscate starlike in D.
Corollary 2.4. If b, p, c ∈C are such that (2.3) √ 3|κ −3| + |c| 4 < s 9 8 + 1 √ 2, then the function zup(z) is lemniscate starlike in D.
Corollary 2.5.
Corollary 2.5. Let p ∈C. For the function Jp(z1/2) = 2pΓ(p + 1)z−p/2Jp(z1/2), where Jp is the Bessel function as defined in (2.4), the…
Corollary 2.5. Let p ∈C. For the function Jp(z1/2) = 2pΓ(p + 1)z−p/2Jp(z1/2), where Jp is the Bessel function as defined in (2.4), the following holds: (i) If |p −1| √ 3 < q 9 8 + 1 √ 2 −1 4, then Jp(z1/2) is lemniscate convex in D. (ii) If |p −2| √
Theorem 2.6.
Theorem 2.6. Let µ, p ∈R be such that µ ± p is not an odd negative integer. If (2.6) 3µ 2 √ 2 − √ 3
Theorem 2.6. Let µ, p ∈R be such that µ ± p is not an odd negative integer. If (2.6) 3µ 2 √ 2 − √ 3
Theorem 2.7.
Theorem 2.7. If µ, p satisfy (2.6), then (hµ,p ∗f)(z) is lemniscate convex in D and thus the functions A[hµ,p](z) and L[hµ,p](z) are…
Theorem 2.7. If µ, p satisfy (2.6), then (hµ,p ∗f)(z) is lemniscate convex in D and thus the functions A[hµ,p](z) and L[hµ,p](z) are lemniscate convex in D. Now, consider the Alexander transform of the function hµ,p(z) named as fµ,p : D →C by fµ,p(z) := zZ 0 hµ,p(t) t dt. The function fµ,p(z) is analytic in D. Moreover, fµ,p ∈A. As hµ,p(z) satisfies the differential equation (1.5), fµ,p(z) satisfies the differential equation z2f ′′′ µ,p(z) + (µ + 2)zf ′′ µ,p(z) + (µ + 1)2 −p2
Theorem 2.8.
Theorem 2.8. Let µ, p ∈R such that µ ± p is not a negative odd integer. If µ, p satisfy (2.7) 3µ 2 √ 2 − √ 3
Theorem 2.8. Let µ, p ∈R such that µ ± p is not a negative odd integer. If µ, p satisfy (2.7) 3µ 2 √ 2 − √ 3
Theorem 2.9.
Theorem 2.9. Let µ, p ∈C be such that µ ± p is not negative odd integer and Re µ > −1 and satisfy (2.8)
Theorem 2.9. Let µ, p ∈C be such that µ ± p is not negative odd integer and Re µ > −1 and satisfy (2.8)
Lemma 3.1.
Lemma 3.1. [12] Let p ∈H[1, n] with p(z) ̸≡1 and n ≥1. Let Ω⊂C and ψ: D ⊂C3 × D →C satisfy ψ(r, s, t; z) ̸∈Ωwhenever z ∈D, r = √ 2…
Lemma 3.1. [12] Let p ∈H[1, n] with p(z) ̸≡1 and n ≥1. Let Ω⊂C and ψ : D ⊂C3 × D →C satisfy ψ(r, s, t; z) ̸∈Ωwhenever z ∈D, r = √ 2 cos2θeiθ, s = me3iθ/(2 √ 2 cos2θ) and Re((t+s)e−3iθ) ≥3m2/(8 √ 2 cos2θ) where m ≥n ≥1 and −π/4 < θ < π/4. If (p(z), zp′(z), z2p′′(z); z) ∈D for z ∈D and ψ(p(z), zp′(z), z2p′′(z); z) ∈Ω, z ∈D, then p(z) ≺√1 + z. In the case ψ : C2 × D →C, the condition in Lemma 3.1 reduces to ψ(r, s; z) ̸∈Ωwhenever r = √ 2 cos2θeiθ, s = me3iθ/(2
Lemma 3.2.
Lemma 3.2. [1, Theorem 2.9, p. 29] If b, p, c ∈C are such that Re κ > |c|/4 + 1, then Re up(z) > 0 for all z ∈D. Further, if Re κ > |c|/4…
Lemma 3.2. [1, Theorem 2.9, p. 29] If b, p, c ∈C are such that Re κ > |c|/4 + 1, then Re up(z) > 0 for all z ∈D. Further, if Re κ > |c|/4 and c ̸= 0, then up is univalent in D.
Lemma 3.1
Lemma 3.1, the theorem follows.
Lemma 3.1, the theorem follows.
Lemma 3.1
Lemma 3.1, we have ψ(r, s, t; z) = t + s + (µ + 1) me3iθ 2 √ 2 cos2θ − (µ + 1)2 −p2 4 + z 4 + (µ + 1)2 −p2 4 √
Lemma 3.1, we have ψ(r, s, t; z) = t + s + (µ + 1) me3iθ 2 √ 2 cos2θ − (µ + 1)2 −p2 4 + z 4 + (µ + 1)2 −p2 4 √
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