花柳界 literally means “Flower Willow World/Kingdom.”

In Japanese, this means “The Realm of the Geisha” or “World of the Geisha.” I suppose there is a presumption that the Geisha is surrounded by flowers in their residence. In Chinese and Korean, this pretty much has colloquially come to mean “The Red Light District” or to refer to pimps, prostitutes, and johns as a group.

芸者 is the real basis for the way we spell geisha.

However, there are many more ways to refer to a woman that fills the role that westerners think of when they hear the word geisha.

In Japanese, these characters literally mean “artful person.” But in English, it might be better translated as “a person (woman) highly trained/accomplished in the arts.”

However, my Japanese dictionary says “a singing and dancing girl.”

Many will argue as to whether “geisha” = “prostitute” or not. My Japanese friends seem to have the opinion that a geisha is so highly trained in the art of playing musical instruments and dancing that the fact she might also be a prostitute is secondary to her performance on stage.

芸者 is a “Japanese only” term, they use a slightly different first character to express “geisha” in Chinese. Since this is a Japanese term, I have not included the Chinese version.

芸妓 is how to refer to a geisha that offers “special services.”

Please don't order this, it's only here for reference. 芸妓 is not appropriate for custom calligraphy!

In Japanese, 名妓 means “distinguished/talented/beautiful geisha.”

The meaning in Chinese (and the deeper meaning in Japanese) would be “distinguished/talented/beautiful prostitute.”

I am not sure that our master calligrapher will even write this, so please note that fact if you decide to place the order. Of course, we'll refund your money if he refuses.

玄人 literally means “Mysterious Person/Woman.”

In Japanese, this is associated with a “geisha,” which matches this definition of a “mysterious woman.” However, this is the colloquial way to say “professional geisha” or “expert prostitute” in Japanese. It, therefore, might not be what you want on your wall.

Notes: This is a "Japanese only" term, though a Chinese person who sees these characters will think of a geisha or an alluring Japanese female musician.
While not often seen in Korean Hanja, this would mean a "dark woman," as in a woman that you cannot easily see through.

This word has a few possible meanings.

It can refer to a room, hall, or place (a dojo or small theater). The dictionary suggests that in some context, this can refer to a special geisha who is talented in certain musical arts.

武芸者 is the Japanese Kanji title for “Martial Arts Master.” It suggests that you have reached at least the level of black belt and are probably to the level where you are ready to become an instructor.

Please consider carefully where you stand before ordering this phrase on a wall scroll. If you are not a master, this will make you look a bit foolish.

If you want to get this as a gift for your master at the dojo. Try to discreetly make sure this term is used in your school. Different schools and styles of Japanese martial arts use different terms. You may notice in the Romaji that the last two characters romanize as “geisha” which means “person skilled in arts” (what a geisha girl really is). The title here has the character for “martial,” “warrior,” and/or “military” in front of it. Therefore the literal translation is “martial art person.”

These Kanji are valid Chinese characters and Korean Hanja, but this title does not really make sense in Chinese and is not often used in Korean, though a Chinese or Korean would be able to guess the meaning by looking at the first and last characters.