The Shōwa's relationship with Hitler and Mussolini was nonexistent, because they have never even met each other, with the Japanese matters of war handled by the ministers and military leaders. The Emperor himself didn't have too much impact on the war effort. The Japanese diplomat Hitler met multiple times was Oshima Hiroshi, and we all know how Hitler viewed Japanese people, even though officially they were accepted as "honorary aryans".
The other, more interesting matter is Hitler's and Mussolini's relationship. Hitler always admired Mussolini and his ideology, he acknowledged it himself that Italian fascism was the "elder brother" of German nazism. Hitler was amazed by Mussolini's March on Rome in 1922, it was basically what inspired his failed coup, in Munich, 1923. Mussolini also provided financial support for the NSDAP, and allowed the SA to train with his own paramilitary brigade, the blackshirts. But not everything is as easy to explain as this. When Hitler came into power in 1933, Mussolini viewed it as a great accomplishment for his own ideology too, but the two nations never had similar interests. By the late 1930s, Germany and Italy had become military allies. Their priorities, however, lay with their own national interests, rather than supporting the interests or ambitions of another country. The union between Nazi Germany and fascist Italy became a marriage of convenience and expedience rather than a firm alliance of sister states. But in private, Mussolini was scornful of Hitler's party and ideology. He viewed the Mein Kampf as 'boring', and Hitler's ideas and theories as 'coarse' and 'simplistic'. They first met in 1934 in Venice, where Mussolini tried to speak German, and refused to use a translator, but he had trouble understanding Hitler's Austrian accent. He was also subjected to the Führer's long monologues, which he disliked greatly. After the meeting, they both thought less of each other, but the public propaganda suggested a strong bond of friendship between the two countries and leaders. Another conflict of interest was their racial views. Hitler considered white Europeans to be the supreme race, most importantly people of Germanic descent (Germans, Danes, Swedes, Norwegians), and he greatly despised Slavic people, Jews, Gypsies and homosexuals, while Mussolini was an Italian nationalist, he didn't consider races as an important part of his ideology. The Italian leader expressed “pity” for the racial views being expressed by the Nazis, “the descendants of those who were illiterate when Rome had Caesar, Virgil and Augustus.”
Despite their personal differences, they did manage to co-operate. Hitler offered support to Italy when they were facing difficulties invading Abyssinia (Ethiopia). In 1937, Mussolini paid a state visit to Germany, and he was thoroughly impressed by the German military equipment, and he signed the Anti-Comintern Pact two months later. German-Italian relations were also boosted by their joint involvement in the Spanish Civil War. Hitler's influence on Mussolini became evident in his 'Manifesto of Race' (1938) when he stripped Italian Jews of their citizenship. In 1939, the two nations signed the Pact of Steel, which was a military pact, and they formed the Berlin-Rome Axis. When WWII started, Mussolini told Hitler Italy wouldn't be ready for war until late 1942, and he didn't militarily support the German invasion of Poland. During the war, Hitler's distrust and annoyance with the Italian military leadership became more evident, (failing to conquer Greece, and failures in North Africa), and in 1943 he ordered his troops to invade Italy, and to install a pro-German government.
Source: 'Hitler and Mussolini' - Jennifer Llewelynn, Jim Southey, Steve Thompson