A class of analytic functions related to the generalized Marcum Q-function and its geometric properties

Lemma 2.1 For min(x,ν) > 0 the following holds: xνe– ν ν+1 x ν ≤γ (ν,x) ≤xν(1 + νe–x) ν(ν + 1) . (2.1)

Theorem 3.37]: (b – a)𝒜 (︁ φ(f ) )︁ ≤ (︁ b – 𝒜(f ) )︁ φ(a) + (︁𝒜 (︁ φ(f ) )︁ – a

Lemma 2.4 [14] If f (z) = z + ∞ ∑︂ k=2 akzk, with ak ≥0 (∀k ≥2) is analytic in 𝔻. Also, if the sequences (kak)k≥1 and (kak – (k + 1)ak+1)k≥1 both are decreasing, then f is starlike in 𝔻. 3 The first set of main results The main focus of this section is to establish sufficient conditions for the function 𝒬ν(z;b) defined by (1.11) to be close-to-convex with respect to some known functions, starlike and starlike of order α. Our first main result in this section reads as follows.

Theorem 3.1 The hollowing assertions hold true: (a). If the parameters b,ν > 0, satisfy the following inequality 2bνe–b ≥νγ (ν,b), (3.1) then the function 𝒬ν(z;b) is close-to-convex with respect to the function –log(1 – z), and consequently it is univalent in 𝔻. (b). Let ν > 0 and b ∈(0,b1] where b1 ≈1.05108··· is the root of the equation 16e– x 2 – 9x = 0. If the inequality (3.1) holds, then the function 𝒢ν(z;b), defined by 𝒢ν(z;b) = z(𝒬ν(z;b))′, z ∈𝔻, (3.2) is close-to-convex with respect to th

Theorem 3.2 Assume that b ∈(0,1] and ν > 0. If the following inequality 4bνe–b ≥3νγ (ν,b), is valid, then the function 𝒬ν(z;b) is starlike in 𝔻.

Corollary 3.3 If b ∈[0,b2), where b2 ≈0.550200 is the unique positive root of the equation 4xe–x + 3e–x – 3 = 0, then the function 𝒬1(z;b) is starlike in 𝔻. Example 3.4 The function 𝒬1(z; 1 2) is starlike in 𝔻. See Fig. 1.

Theorem 3.6 Let b,ν > 0 such that ν(ν + 1)γ (ν,b) > bν(eb – 1), and there exists a real α such that 0 ≤α < 1 – bν eb ν(ν + 1)γ (ν,b) – bν(︁ eb – 1 )︁. (3.9) Then, the function 𝒬ν(z;b) is starlike of order α in 𝔻.

Corollary 3.7 Suppose that b,ν > 0 which satisfies the following inequality: bν(2eb – 1) < ν(ν + 1)γ (ν,b). Then, the function 𝒬ν(z;b) is starlike in 𝔻. Upon setting α = 1 2 in Theorem 3.6, we compute the following result.

Corollary 3.8 Suppose that b,ν > 0 which satisfies the following inequality: bν(3eb – 1) < ν(ν + 1)γ (ν,b). Then the function 𝒬ν(z;b) is starlike of order 1 2.

Theorem 3.9 Let ν,b > 0 such as bν(eb – 1) ≤ν(ν + 1)γ (ν,b). Then the function 𝒬ν(z;b) is starlike in 𝔻1 2 .

Corollary 3.10 Let b ∈(0,b3) where b3 ≈0.852605 is the unique positive root of the equa- tion xex – x – 2 + 2e–x = 0, then the function 𝒬1(z;b) is starlike in 𝔻1 2 . Example 3.11 The function 𝒬1(z;85/100) is starlike in 𝔻1 2 , see Fig. 2. 4 The second set of main results Our aim in this section is to find sufficient conditions for the function 𝒬ν(z;b) defined by (1.11) to be convex in 𝔻(resp. in 𝔻1 2 ) and convex of order α.

Theorem 4.1 Let ν > 0 and b ∈(0,b1]. If the inequality (3.1) holds, then the function 𝒬ν(z;b) is convex in 𝔻.

Corollary 4.2 If b ∈(0,b1], then the function 𝒬1(z;b) is convex in 𝔻. Example 4.3 The function 𝒬1(z;105/100) is convex in 𝔻. See Fig. 3.

Theorem 4.5 Let b > 0,ν ≥1 and α ≥0 such that 0 ≤α ≤1 – 2bν(eb – 1) ν(ν + 1)γ (ν,b) – 2bν(eb – 1). Then, the function 𝒬ν(z;b) is convex of order α in 𝔻.

Corollary 4.6 Suppose that b > 0 and ν ≥1 such that 4bν(eb – 1) ≤ν(ν + 1)γ (ν,b). Then, the function 𝒬ν(z;b) is convex in 𝔻.

Theorem 4.7 Let ν,b > 0. If the following inequality holds true: bν–1(1 – e–b)[(b + 1)eb – 1] ≤ν(ν + 1)γ (ν,b), then the function 𝒬ν(z;b) is convex in 𝔻1 2 .

Corollary 4.8 Let b ∈(0,b4) where b4 ≈0.617642 is the unique positive root of the equa- tion 3 – (x + 1)ex = 0, then the function 𝒬1(z;b) is convex in 𝔻1 2 . Example 4.9 The function 𝒬1(z;6/10) is convex in 𝔻1 2 . See Fig. 4. 5 Conclusion In our present paper, we have established new geometric properties such as starlikeness, convexity and close-to-convexity inside the unit disk 𝔻of a class of analytic functions related to the generalized Marcum Q-function Qν(a,b), consult (1.2). Moreover, sever